CHEM 105: Introduction to General, Organic, and Biological Chemistry PDF

CHEM 105: INTRODUCTION TO GENERAL, ORGANIC, AND BIOLOGICAL CHEMISTRY FALL 22 Meghan M Klems Widener University CHEM 105: Introduction to General, Organic, and Biological Chemistry Fall 22 This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.or...

CHEM 105: INTRODUCTION TO GENERAL, ORGANIC, AND BIOLOGICAL CHEMISTRY FALL 22 Meghan M Klems Widener University CHEM 105: Introduction to General, Organic, and Biological Chemistry Fall 22 This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the hundreds of other texts available within this powerful platform, it is freely available for reading, printing and "consuming." Most, but not all, pages in the library have licenses that may allow individuals to make changes, save, and print this book. Carefully consult the applicable license(s) before pursuing such effects. Instructors can adopt existing LibreTexts texts or Remix them to quickly build course-specific resources to meet the needs of their students. Unlike traditional textbooks, LibreTexts’ web based origins allow powerful integration of advanced features and new technologies to support learning. 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This text was compiled on 01/13/2024 TABLE OF CONTENTS Licensing 1: What is Chemistry? 1.1: Chemistry and Chemicals 1.2: Chemicals Compose Ordinary Things 1.3: The Scientific Method - How Chemists Think 1.4: Hypothesis, Theories, and Laws 1.E: Exercises 2: Measurements and Problem Solving 2.1: Taking Measurements 2.2: Scientific Notation - Writing Large and Small Numbers 2.3: Significant Figures - Writing Numbers to Reflect Precision 2.4: Significant Figures in Calculations 2.5: The Basic Units of Measurement 2.6: Problem Solving and Unit Conversions 2.7: Solving Multi-step Conversion Problems 2.8: Density 3: Matter and Energy 3.1: In Your Room 3.2: Classifying Matter According to Its State—Solid, Liquid, and Gas 3.3: Classifying Matter According to Its Composition 3.4: Differences in Matter- Physical and Chemical Properties 3.5: Changes in Matter - Physical and Chemical Changes 3.6: Conservation of Mass - There is No New Matter 3.7: Energy 3.8: Temperature - Random Motion of Molecules and Atoms 3.9: Energy and Nutrition 3.E: Matter and Energy (Exercises) 4: Elements, Atoms, and the Periodic Table 4.1: Prelude to Elements, Atoms, and the Periodic Table 4.2: The Elements 4.3: Atomic Theory 4.4: The Structure of Atoms 4.5: Nuclei of Atoms 4.6: Atomic Masses 4.7: Arrangements of Electrons 4.8: The Periodic Table 4.E: Elements, Atoms, and the Periodic Table (Exercises) 4.S: Elements, Atoms, and the Periodic Table (Summary) 1 https://chem.libretexts.org/@go/page/402594 5: Ionic Bonding and Simple Ionic Compounds 5.1: Prelude to Ionic Bonding and Simple Ionic Compounds 5.2: Two Types of Bonding 5.3: Ions 5.4: Formulas for Ionic Compounds 5.5: Ionic Nomenclature 5.6: Characteristics of Ionic Compounds 5.E: Ionic Bonding and Simple Ionic Compounds (Exercises) 5.S: Ionic Bonding and Simple Ionic Compounds (Summary) 6: Covalent Bonding and Simple Molecular Compounds 6.1: Prelude to Covalent Bonding and Simple Molecular Compounds 6.2: Covalent Bonds 6.3: Covalent Compounds - Formulas and Names 6.4: Drawing Lewis Structures 6.5: Characteristics of Covalent Bonds 6.6: Characteristics of Molecules 6.E: Covalent Bonding and Simple Molecular Compounds (Exercises) 6.S: Covalent Bonding and Simple Molecular Compounds (Summary) 7: Introduction to Chemical Reactions 7.1: Prelude to Introduction to Chemical Reactions 7.2: Formula Mass and the Mole Concept 7.3: The Law of Conservation of Matter 7.4: Chemical Equations 7.5: Quantitative Relationships Based on Chemical Equations 7.6: Some Types of Chemical Reactions 7.E: Introduction to Chemical Reactions (Exercises) 7.S: Introduction to Chemical Reactions (Summary) 8: Solutions 8.1: Prelude to Solutions 8.2: Solutions 8.3: Concentration 8.4: The Dissolution Process 8.5: Properties of Solutions 8.6: Chemical Equilibrium 8.7: Le Chatelier's Principle 8.8: Osmosis and Diffusion 8.E: Solutions (Exercises) 8.S: Solutions (Summary) 9: Acids and Bases 9.1: Prelude to Acids and Bases 9.2: Arrhenius Definition of Acids and Bases 9.3: Brønsted-Lowry Definition of Acids and Bases 9.4: Water - Both an Acid and a Base 9.5: The Strengths of Acids and Bases 9.6: Buffers 9.E: Acids and Bases (Exercises) 2 https://chem.libretexts.org/@go/page/402594  9.S: Acids and Bases (Summary) 10: Organic Chemistry - Alkanes and Halogenated Hydrocarbons 10.1: Prelude to Organic Chemistry - Alkanes and Halogenated Hydrocarbons 10.2: Organic Chemistry 10.3: Structures and Names of Alkanes 10.4: Branched-Chain Alkanes 10.5: Condensed Structural and Line-Angle Formulas 10.6: IUPAC Nomenclature 10.7: Physical Properties of Alkanes 10.8: Chemical Properties of Alkanes 10.9: Halogenated Hydrocarbons 10.10: Cycloalkanes 10.E: Organic Chemistry- Alkanes and Halogenated Hydrocarbons (Exercises) 10.S: Organic Chemistry- Alkanes and Halogenated Hydrocarbons (Summary) 11: Unsaturated and Aromatic Hydrocarbons 11.1: Prelude to Unsaturated and Aromatic Hydrocarbons 11.2: Alkenes- Structures and Names 11.3: Cis-Trans Isomers (Geometric Isomers) 11.4: Physical Properties of Alkenes 11.5: Chemical Properties of Alkenes 11.6: Polymers 11.7: Alkynes 11.8: Aromatic Compounds- Benzene 11.9: Structure and Nomenclature of Aromatic Compounds 11.E: Unsaturated and Aromatic Hydrocarbons (Exercises) 11.S: Unsaturated and Aromatic Hydrocarbons (Summary) 12: Organic Compounds of Oxygen 12.1: Prelude to Organic Compounds of Oxygen 12.2: Organic Compounds with Functional Groups 12.3: Alcohols - Nomenclature and Classification 12.4: Physical Properties of Alcohols 12.5: Reactions that Form Alcohols 12.6: Reactions of Alcohols 12.7: Glycols and Glycerol 12.8: Phenols 12.9: Ethers 12.10: Aldehydes and Ketones- Structure and Names 12.11: Properties of Aldehydes and Ketones 12.12: Organic Sulfur Compounds 12.E: Organic Compounds of Oxygen (Exercises) 12.S: Organic Compounds of Oxygen (Summary) 13: Organic Acids and Bases and Some of Their Derivatives 13.1: Prelude to Organic Acids and Bases and Some of Their Derivatives 13.2: Carboxylic Acids - Structures and Names 13.3: The Formation of Carboxylic Acids 13.4: Physical Properties of Carboxylic Acids 3 https://chem.libretexts.org/@go/page/402594  13.5: Chemical Properties of Carboxylic Acids- Ionization and Neutralization 13.6: Esters - Structures and Names 13.7: Physical Properties of Esters 13.8: Preparation of Esters 13.9: Hydrolysis of Esters 13.10: Esters of Phosphoric Acid 13.11: Amines - Structures and Names 13.12: Physical Properties of Amines 13.13: Amines as Bases 13.14: Amides- Structures and Names 13.15: Physical Properties of Amides 13.16: Formation of Amides 13.17: Chemical Properties of Amides- Hydrolysis 13.S: Organic Acids and Bases and Some of Their Derivatives (Summary) 14: Carbohydrates 14.1: Prelude to Carbohydrates 14.2: Carbohydrates 14.3: Classes of Monosaccharides 14.4: Important Hexoses 14.5: Cyclic Structures of Monosaccharides 14.6: Properties of Monosaccharides 14.7: Disaccharides 14.8: Polysaccharides 14.S: Carbohydrates (Summary) 15: Amino Acids, Proteins, and Enzymes 15.1: Prelude to Amino Acids, Proteins, and Enzymes 15.2: Properties of Amino Acids 15.3: Reactions of Amino Acids 15.4: Peptides 15.5: Proteins 15.6: Enzymes 15.7: Enzyme Action 15.8: Enzyme Activity 15.9: Enzyme Inhibition 15.10: Enzyme Cofactors and Vitamins 15.E: Amino Acids, Proteins, and Enzymes (Exercises) 15.S: Amino Acids, Proteins, and Enzymes (Summary) 16: Lipids 16.1: Prelude to Lipids 16.2: Fatty Acids 16.3: Fats and Oils 16.4: Membranes and Membrane Lipids 16.5: Steroids 16.E: Exercises 16.S: Lipids (Summary) 4 https://chem.libretexts.org/@go/page/402594 Index Glossary Detailed Licensing 5 https://chem.libretexts.org/@go/page/402594 Licensing A detailed breakdown of this resource's licensing can be found in Back Matter/Detailed Licensing. 1 https://chem.libretexts.org/@go/page/428639  CHAPTER OVERVIEW 1: What is Chemistry? 1.1: Chemistry and Chemicals 1.2: Chemicals Compose Ordinary Things 1.3: The Scientific Method - How Chemists Think 1.4: Hypothesis, Theories, and Laws 1.E: Exercises 1: What is Chemistry? is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts. 1 1.1: Chemistry and Chemicals  Learning Objectives To recognize the breadth, depth, and scope of chemistry. Define chemistry in relation to other sciences. Identify the main disciplines of chemistry. Define Chemical. Chemistry is the study of matter—what it consists of, what its properties are, and how it changes. Matter is anything that has mass and takes up space—that is, anything that is physically real. Some things are easily identified as matter—the screen on which you are reading this book, for example. Others are not so obvious. Because we move so easily through air, we sometimes forget that it, too, is matter. Because of this, chemistry is a science that has its fingers in just about everything. Being able to describe the ingredients in a cake and how they change when the cake is baked, for example, is chemistry! Chemistry is one of the many branches of science. Science is the process by which we learn about the natural universe by observing, testing, and then generating models that explain our observations. Because the physical universe is so vast, there are many different branches of science (Figure 1.1.1). Thus, chemistry is the study of matter, biology is the study of living things, and geology is the study of rocks and the earth. Mathematics is the language of science, and we will use it to communicate some of the ideas of chemistry. Figure 1.1.1 : The Relationships between Some of the Major Branches of Science. Chemistry lies more or less in the middle, which emphasizes its importance to many branches of science. Although we divide science into different fields, there is much overlap among them. For example, some biologists and chemists work in both fields so much that their work is called biochemistry. Similarly, geology and chemistry overlap in the field called geochemistry. Figure 1.1.1 shows how many of the individual fields of science are related. At some level, all of these fields depend on matter because they all involve "stuff"; because of this, chemistry has been called the "central science", linking them all together. There are many other fields of science, in addition to the ones (biology, medicine, etc.) listed here. 1.1.1 https://chem.libretexts.org/@go/page/402414   Exercise 1.1.1 Which fields of study are branches of science? a. physiology (the study of the function of an animal’s or a plant’s body) b. geophysics c. agriculture d. politics Answer a: yes Answer b: yes Answer c: yes Answer d: no Areas of Chemistry The study of modern chemistry has many branches, but can generally be broken down into five main disciplines, or areas of study: Physical chemistry: Physical chemistry is the study of macroscopic properties, atomic properties, and phenomena in chemical systems. A physical chemist may study such things as the rates of chemical reactions, the energy transfers that occur in reactions, or the physical structure of materials at the molecular level. Organic chemistry: Organic chemistry is the study of chemicals containing carbon. Carbon is one of the most abundant elements on Earth and is capable of forming a tremendously vast number of chemicals (over twenty million so far). Most of the chemicals found in all living organisms are based on carbon. Inorganic chemistry: Inorganic chemistry is the study of chemicals that, in general, are not primarily based on carbon. Inorganic chemicals are commonly found in rocks and minerals. One current important area of inorganic chemistry deals with the design and properties of materials involved in energy and information technology. Analytical chemistry: Analytical chemistry is the study of the composition of matter. It focuses on separating, identifying, and quantifying chemicals in samples of matter. An analytical chemist may use complex instruments to analyze an unknown material in order to determine its various components. Biochemistry: Biochemistry is the study of chemical processes that occur in living things. Research may cover anything from basic cellular processes up to understanding disease states so that better treatments can be developed. Figure 1.1.2 : (left) Measurement of trace metals using atomic spectroscopy. (right) Measurement of hormone concentrations. In practice, chemical research is often not limited to just one of the five major disciplines. A particular chemist may use biochemistry to isolate a particular chemical found in the human body such as hemoglobin, the oxygen carrying component of red blood cells. He or she may then proceed to analyze the hemoglobin using methods that would pertain to the areas of physical or analytical chemistry. Many chemists specialize in areas that are combinations of the main disciplines, such as bioinorganic chemistry or physical organic chemistry. 1.1.2 https://chem.libretexts.org/@go/page/402414 What is a Chemical? A chemical is a substance that always has the same composition and properties wherever it is found. Since matter is anything that has mass and volume, chemicals are the substances that make up all matter including the air we breathe, the water we drink, or the desk that you're sitting. If you think about it, everything is chemicals, and you use some pretty common ones every day as seen below. Summary Chemistry is the study of matter and the changes it undergoes and considers both macroscopic and microscopic information. Matter is anything that has mass and occupies space. The five main disciplines of chemistry are physical chemistry, organic chemistry, inorganic chemistry, analytical chemistry and biochemistry. A chemical is a substance that always has the same omposition and properties wherever it is found. 1.1: Chemistry and Chemicals is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 1.7: The Scope of Chemistry by Henry Agnew, Marisa Alviar-Agnew is licensed CK-12. Original source: https://www.ck12.org/c/chemistry/. 1.1.3 https://chem.libretexts.org/@go/page/402414 1.2: Chemicals Compose Ordinary Things Chemistry is the branch of science dealing with the structure, composition, properties, and the reactive characteristics of matter. Matter is anything that has mass and occupies space. Thus, chemistry is the study of literally everything around us—the liquids that we drink, the gases we breathe, the composition of everything from the plastic case on your phone to the earth beneath your feet. Moreover, chemistry is the study of the transformation of matter. In this chapter, we will discuss some of the properties of matter and how chemists measure those properties. We will introduce some of the vocabulary that is used throughout chemistry and the other physical sciences. Let’s begin with matter. Matter is defined as any substance that has mass and occupies space, also known as volume. It is important to distinguish here between weight and mass. Weight is the result of the pull of gravity on an object. On the Moon, an object will weigh less than the same object on Earth because the pull of gravity is less on the Moon. The mass of an object, however, is an inherent property of that object and does not change, regardless of location, gravitational pull, or anything else. It is a property that is solely dependent on the quantity of matter within the object. Contemporary theories suggests that matter is composed of atoms. Atoms themselves are constructed from neutrons, protons and electrons, along with an ever-increasing array of other subatomic particles. We will focus on the neutron, a particle having no charge; the proton, which carries a positive charge; and the electron, which has a negative charge. Atoms are incredibly small. To give you an idea of the size of an atom, a single copper penny contains approximately 28,000,000,000,000,000,000,000 atoms (that’s 28 sextillion). Because atoms and subatomic particles are so small, their mass is not readily measured using pounds, ounces, grams or any other scale that we would use on larger objects. Instead, the mass of atoms and subatomic particles is measured using atomic mass units (abbreviated amu). The atomic mass unit is based on a scale that relates the mass of different types of atoms to each other (using the most common form of the element carbon as a standard). The amu scale gives us a convenient means to describe the masses of individual atoms and to do quantitative measurements concerning atoms and their reactions. Within an atom, the neutron and proton both have a mass of one amu; the electron has a much smaller mass (about 0.0005 amu). Figure 1.2.1 : Atoms are incredible small. To give you an idea of the size of an atom, a single copper penny contains approximately 28,000,000,000,000,000,000,000 atoms (that’s 28 sextillion). Atomic theory places the neutron and the proton in the center of the atom in the nucleus. In an atom, the nucleus is very small, very dense, carries a positive charge (from the protons) and contains virtually all of the mass of the atom. Electrons are placed in a diffuse cloud surrounding the nucleus. The electron cloud carries a net negative charge (from the charge on the electrons) and in a neutral atom there are always as many electrons in this cloud as there are protons in the nucleus (the positive charges in the nucleus are balanced by the negative charges of the electrons, making the atom neutral). An atom is characterized by the number of neutrons, protons and electrons that it possesses. Today, we recognize at least 116 different types of atoms, each type having a different number of protons in its nucleus. These different types of atoms are called elements. The neutral element hydrogen (the lightest element) will always have one proton in its nucleus and one electron in the cloud surrounding the nucleus. The element helium will always have two protons in its nucleus. It is the number of protons in the nucleus of an atom that defines the identity of an element. Elements can, however, have differing numbers of neutrons in their nucleus. For example, stable helium nuclei exist that contain one, or two neutrons (but they all have two protons). These different types of helium atoms have different masses (3 or 4 amu) and they are called isotopes. For any given isotope, the sum of the numbers of protons and neutrons in the nucleus is called the mass number. All elements exist as a collection of isotopes, and the mass of an element that we use in chemistry, the atomic mass, is the average of the masses of these isotopes. For helium, there is approximately one isotope of Helium-3 for every one million isotopes of Helium-4, hence the average atomic mass is very close to 4 (4.002602). As different elements were discovered and named, abbreviations of their names were developed to allow for a convenient chemical shorthand. The abbreviation for an element is called its chemical symbol. A chemical symbol consists of one or two letters, and the 1.2.1 https://chem.libretexts.org/@go/page/402415 relationship between the symbol and the name of the element is generally apparent. Thus helium has the chemical symbol He, nitrogen is N, and lithium is Li. Sometimes the symbol is less apparent but is decipherable; magnesium is Mg, strontium is Sr, and manganese is Mn. Symbols for elements that have been known since ancient times, however, are often based on Latin or Greek names and appear somewhat obscure from their modern English names. For example, copper is Cu (from cuprum), silver is Ag (from argentum), gold is Au (from aurum), and iron is Fe (from ferrum). Throughout your study of chemistry, you will routinely use chemical symbols and it is important that you begin the process of learning the names and chemical symbols for the common elements. By the time you complete General Chemistry, you will find that you are adept at naming and identifying virtually all of the 116 known elements. Table 1.2.1 contains a starter list of common elements that you should begin learning now! Table 1.2.1 : Names and Chemical Symbols for Common Elements Element Chemical Symbol Element Chemical Symbol Hydrogen H Phosphorus P Helium He Sulfur S Lithium Li Chlorine Cl Beryllium Be Argon Ar Boron B Potassium K Carbon C Calcium Ca Nitrogen N Iron Fe Oxygen O Copper Cu Fluorine F Zinc Zn Neon Ne Bromine Br Sodium Na Silver Ag Magnesium Mg Iodine I Aluminum Al Gold Au Silicon Si Lead Pb The chemical symbol for an element is often combined with information regarding the number of protons and neutrons in a particular isotope of that atom to give the atomic symbol. To write an atomic symbol, begin with the chemical symbol, then write the atomic number for the element (the number of protons in the nucleus) as a subscript, preceding the chemical symbol. Directly above this, as a superscript, write the mass number for the isotope, that is, the total number of protons and neutrons in the nucleus. Thus, for helium, the atomic number is 2 and there are two neutrons in the nucleus for the most common isotope, making the atomic symbol He. In the definition of the atomic mass unit, the “most common isotope of carbon”, C, is defined as having a 4 2 12 6 mass of exactly 12 amu and the atomic masses of the remaining elements are based on their masses relative to this isotope. Chlorine (chemical symbol Cl) consists of two major isotopes, one with 18 neutrons (the most common, comprising 75.77% of natural chlorine atoms) and one with 20 neutrons (the remaining 24.23%). The atomic number of chlorine is 17 (it has 17 protons in its nucleus), therefore the chemical symbols for the two isotopes are Cl and Cl. 35 17 37 17 When data is available regarding the natural abundance of various isotopes of an element, it is simple to calculate the average atomic mass. In the example above, Cl was the most common isotope with an abundance of 75.77% and Cl had an abundance 35 17 37 17 of the remaining 24.23%. To calculate the average mass, first convert the percentages into fractions; that is, simply divide them by 100. Now, chlorine-35 represents a fraction of natural chlorine of 0.7577 and has a mass of 35 (the mass number). Multiplying these, we get (0.7577 × 35) = 26.51. To this, we need to add the fraction representing chlorine-37, or (0.2423 × 37) = 8.965; adding, (26.51 + 8.965) = 35.48, which is the weighted average atomic mass for chlorine. Whenever we do mass calculations involving elements or compounds (combinations of elements), we always need to use average atomic masses. Contributions & Attributions Paul R. Young, Professor of Chemistry, University of Illinois at Chicago, Wiki: AskTheNerd; PRY askthenerd.com - pyoung uic.edu; ChemistryOnline.com 1.2.2 https://chem.libretexts.org/@go/page/402415 1.2: Chemicals Compose Ordinary Things is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 1.2: Chemicals Compose Ordinary Things by Henry Agnew, Marisa Alviar-Agnew is licensed CK-12. Original source: https://www.ck12.org/c/chemistry/. 1.2.3 https://chem.libretexts.org/@go/page/402415 1.3: The Scientific Method - How Chemists Think  Learning Objectives Identify the components of the scientific method. Scientists search for answers to questions and solutions to problems by using a procedure called the scientific method. This procedure consists of making observations, formulating hypotheses, and designing experiments; which leads to additional observations, hypotheses, and experiments in repeated cycles (Figure 1.3.1). Figure 1.3.1 : The Steps in the Scientific Method. Step 1: Make observations Observations can be qualitative or quantitative. Qualitative observations describe properties or occurrences in ways that do not rely on numbers. Examples of qualitative observations include the following: "the outside air temperature is cooler during the winter season," "table salt is a crystalline solid," "sulfur crystals are yellow," and "dissolving a penny in dilute nitric acid forms a blue solution and a brown gas." Quantitative observations are measurements, which by definition consist of both a number and a unit. Examples of quantitative observations include the following: "the melting point of crystalline sulfur is 115.21° Celsius," and "35.9 grams of table salt—the chemical name of which is sodium chloride—dissolve in 100 grams of water at 20° Celsius." For the question of the dinosaurs’ extinction, the initial observation was quantitative: iridium concentrations in sediments dating to 66 million years ago were 20–160 times higher than normal. Step 2: Formulate a hypothesis After deciding to learn more about an observation or a set of observations, scientists generally begin an investigation by forming a hypothesis, a tentative explanation for the observation(s). The hypothesis may not be correct, but it puts the scientist’s understanding of the system being studied into a form that can be tested. For example, the observation that we experience alternating periods of light and darkness corresponding to observed movements of the sun, moon, clouds, and shadows is consistent with either one of two hypotheses: a. Earth rotates on its axis every 24 hours, alternately exposing one side to the sun. b. The sun revolves around Earth every 24 hours. Suitable experiments can be designed to choose between these two alternatives. For the disappearance of the dinosaurs, the hypothesis was that the impact of a large extraterrestrial object caused their extinction. Unfortunately (or perhaps fortunately), this hypothesis does not lend itself to direct testing by any obvious experiment, but scientists can collect additional data that either support or refute it. Step 3: Design and perform experiments After a hypothesis has been formed, scientists conduct experiments to test its validity. Experiments are systematic observations or measurements, preferably made under controlled conditions—that is—under conditions in which a single variable changes. 1.3.1 https://chem.libretexts.org/@go/page/402416 Step 4: Accept or modify the hypothesis A properly designed and executed experiment enables a scientist to determine whether or not the original hypothesis is valid. If the hypothesis is valid, the scientist can proceed to step 5. In other cases, experiments often demonstrate that the hypothesis is incorrect or that it must be modified and requires further experimentation. Step 5: Development into a law and/or theory More experimental data are then collected and analyzed, at which point a scientist may begin to think that the results are sufficiently reproducible (i.e., dependable) to merit being summarized in a law, a verbal or mathematical description of a phenomenon that allows for general predictions. A law simply states what happens; it does not address the question of why. One example of a law, the law of definite proportions, which was discovered by the French scientist Joseph Proust (1754–1826), states that a chemical substance always contains the same proportions of elements by mass. Thus, sodium chloride (table salt) always contains the same proportion by mass of sodium to chlorine, in this case 39.34% sodium and 60.66% chlorine by mass, and sucrose (table sugar) is always 42.11% carbon, 6.48% hydrogen, and 51.41% oxygen by mass. Whereas a law states only what happens, a theory attempts to explain why nature behaves as it does. Laws are unlikely to change greatly over time unless a major experimental error is discovered. In contrast, a theory, by definition, is incomplete and imperfect, evolving with time to explain new facts as they are discovered. Because scientists can enter the cycle shown in Figure 1.3.1 at any point, the actual application of the scientific method to different topics can take many different forms. For example, a scientist may start with a hypothesis formed by reading about work done by others in the field, rather than by making direct observations.  Example 1.3.1 Classify each statement as a law, a theory, an experiment, a hypothesis, an observation. a. Ice always floats on liquid water. b. Birds evolved from dinosaurs. c. Hot air is less dense than cold air, probably because the components of hot air are moving more rapidly. d. When 10 g of ice were added to 100 mL of water at 25°C, the temperature of the water decreased to 15.5°C after the ice melted. e. The ingredients of Ivory soap were analyzed to see whether it really is 99.44% pure, as advertised. Solution a. This is a general statement of a relationship between the properties of liquid and solid water, so it is a law. b. This is a possible explanation for the origin of birds, so it is a hypothesis. c. This is a statement that tries to explain the relationship between the temperature and the density of air based on fundamental principles, so it is a theory. d. The temperature is measured before and after a change is made in a system, so these are observations. e. This is an analysis designed to test a hypothesis (in this case, the manufacturer’s claim of purity), so it is an experiment.  Exercise 1.3.1 Classify each statement as a law, a theory, an experiment, a hypothesis, a qualitative observation, or a quantitative observation. a. Measured amounts of acid were added to a Rolaids tablet to see whether it really “consumes 47 times its weight in excess stomach acid.” b. Heat always flows from hot objects to cooler ones, not in the opposite direction. c. The universe was formed by a massive explosion that propelled matter into a vacuum. d. Michael Jordan is the greatest pure shooter to ever play professional basketball. e. Limestone is relatively insoluble in water, but dissolves readily in dilute acid with the evolution of a gas. Answer a experiment Answer b 1.3.2 https://chem.libretexts.org/@go/page/402416  law Answer c theory Answer d hypothesis Answer e observation Summary The scientific method is a method of investigation involving experimentation and observation to acquire new knowledge, solve problems, and answer questions. The key steps in the scientific method include the following: Step 1: Make observations. Step 2: Formulate a hypothesis. Step 3: Test the hypothesis through experimentation. Step 4: Accept or modify the hypothesis. Step 5: Develop into a law and/or a theory. Contributions & Attributions Wikipedia 1.3: The Scientific Method - How Chemists Think is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar- Agnew & Henry Agnew. 1.3: The Scientific Method - How Chemists Think by Henry Agnew, Marisa Alviar-Agnew is licensed CK-12. Original source: https://www.ck12.org/c/chemistry/. 1.3.3 https://chem.libretexts.org/@go/page/402416 1.4: Hypothesis, Theories, and Laws  Learning Objectives Describe the difference between hypothesis and theory as scientific terms. Describe the difference between a theory and scientific law. Although many have taken science classes throughout the course of their studies, people often have incorrect or misleading ideas about some of the most important and basic principles in science. Most students have heard of hypotheses, theories, and laws, but what do these terms really mean? Prior to reading this section, consider what you have learned about these terms before. What do these terms mean to you? What do you read that contradicts or supports what you thought? What is a Fact? A fact is a basic statement established by experiment or observation. All facts are true under the specific conditions of the observation. What is a Hypothesis? One of the most common terms used in science classes is a "hypothesis". The word can have many different definitions, depending on the context in which it is being used: An educated guess: a scientific hypothesis provides a suggested solution based on evidence. Prediction: if you have ever carried out a science experiment, you probably made this type of hypothesis when you predicted the outcome of your experiment. Tentative or proposed explanation: hypotheses can be suggestions about why something is observed. In order for it to be scientific, however, a scientist must be able to test the explanation to see if it works and if it is able to correctly predict what will happen in a situation. For example, "if my hypothesis is correct, we should see ___ result when we perform ___ test." A hypothesis is very tentative; it can be easily changed. What is a Theory? The United States National Academy of Sciences describes what a theory is as follows: "Some scientific explanations are so well established that no new evidence is likely to alter them. The explanation becomes a scientific theory. In everyday language a theory means a hunch or speculation. Not so in science. In science, the word theory refers to a comprehensive explanation of an important feature of nature supported by facts gathered over time. Theories also allow scientists to make predictions about as yet unobserved phenomena." "A scientific theory is a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experimentation. Such fact-supported theories are not "guesses" but reliable accounts of the real world. The theory of biological evolution is more than "just a theory." It is as factual an explanation of the universe as the atomic theory of matter (stating that everything is made of atoms) or the germ theory of disease (which states that many diseases are caused by germs). Our understanding of gravity is still a work in progress. But the phenomenon of gravity, like evolution, is an accepted fact. Note some key features of theories that are important to understand from this description: Theories are explanations of natural phenomena. They aren't predictions (although we may use theories to make predictions). They are explanations as to why we observe something. Theories aren't likely to change. They have a large amount of support and are able to satisfactorily explain numerous observations. Theories can, indeed, be facts. Theories can change, but it is a long and difficult process. In order for a theory to change, there must be many observations or pieces of evidence that the theory cannot explain. Theories are not guesses. The phrase "just a theory" has no room in science. To be a scientific theory carries a lot of weight; it is not just one person's idea about something Theories aren't likely to change. 1.4.1 https://chem.libretexts.org/@go/page/402417 What is a Law? Scientific laws are similar to scientific theories in that they are principles that can be used to predict the behavior of the natural world. Both scientific laws and scientific theories are typically well-supported by observations and/or experimental evidence. Usually scientific laws refer to rules for how nature will behave under certain conditions, frequently written as an equation. Scientific theories are more overarching explanations of how nature works and why it exhibits certain characteristics. As a comparison, theories explain why we observe what we do and laws describe what happens. For example, around the year 1800, Jacques Charles and other scientists were working with gases to, among other reasons, improve the design of the hot air balloon. These scientists found, after many, many tests, that certain patterns existed in the observations on gas behavior. If the temperature of the gas is increased, the volume of the gas increased. This is known as a natural law. A law is a relationship that exists between variables in a group of data. Laws describe the patterns we see in large amounts of data, but do not describe why the patterns exist.  Laws vs. Theories A common misconception is that scientific theories are rudimentary ideas that will eventually graduate into scientific laws when enough data and evidence has accumulated. A theory does not change into a scientific law with the accumulation of new or better evidence. Remember, theories are explanations and laws are patterns we see in large amounts of data, frequently written as an equation. A theory will always remain a theory; a law will always remain a law. What’s the difference between a scien… scien… Video 1.4.1: What’s the difference between a scientific law and theory? Summary A hypothesis is a tentative explanation that can be tested by further investigation. A theory is a well-supported explanation of observations. A scientific law is a statement that summarizes the relationship between variables. An experiment is a controlled method of testing a hypothesis. Contributions & Attributions Marisa Alviar-Agnew (Sacramento City College) Henry Agnew (UC Davis) 1.4: Hypothesis, Theories, and Laws is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 1.6: Hypothesis, Theories, and Laws by Henry Agnew, Marisa Alviar-Agnew is licensed CK-12. Original source: https://www.ck12.org/c/chemistry/. 1.4.2 https://chem.libretexts.org/@go/page/402417 1.E: Exercises 1.1: Soda Pop Fizz 1.2: Chemicals Compose Ordinary Things 1.3: All Things Are Made of Atoms and Molecules 1.4: The Scientific Method: How Chemists Think Use the following paragraph to answer the first two questions. In 1928, Sir Alexander Fleming was studying Staphylococcus bacteria growing in culture dishes. He noticed that a mold called Penicillium was also growing in some of the dishes. In Figure 1.13, Petri dish A represents a dish containing only Staphylococcus bacteria. The red dots in dish B represent Penicillium colonies. Fleming noticed that a clear area existed around the mold because all the bacteria grown in this area had died. In the culture dishes without the mold, no clear areas were present. Fleming suggested that the mold was producing a chemical that killed the bacteria. He decided to isolate this substance and test it to see if it would kill bacteria. Fleming grew some Penicillium mold in a nutrient broth. After the mold grew in the broth, he removed all the mold from the broth and added the broth to a culture of bacteria. All the bacteria died. 1. Which of the following statements is a reasonable expression of Fleming’s hypothesis? a. Nutrient broth kills bacteria. b. There are clear areas around the Penicillium mold where Staphylococcus doesn't grow. c. Mold kills bacteria. d. Penicillium mold produces a substance that kills Staphylococcus. e. Without mold in the culture dish, there were no clear areas in the bacteria. 2. Fleming grew Penicillium in broth, then removed the Penicillium and poured the broth into culture dishes containing bacteria to see if the broth would kill the bacteria. What step in the scientific method does this represent? a. Collecting and organizing data b. Making a hypothesis c. Testing a hypothesis by experiment d. Rejecting the old hypothesis and making a new one e. None of these A scientific investigation is NOT valid unless every step in the scientific method is present and carried out in the exact order listed in this chapter. a. True b. False Which of the following words is closest to the same meaning as hypothesis? a. fact b. law c. formula d. suggestion e. conclusion Why do scientists sometimes discard theories? a. the steps in the scientific method were not followed in order b. public opinion disagrees with the theory c. the theory is opposed by the church d. contradictory observations are found e. congress voted against it Gary noticed that two plants which his mother planted on the same day, that were the same size when planted, were different in size after three weeks. Since the larger plant was in the full sun all day and the smaller plant was in the shade of a tree most of the day, Gary believed the sunshine was responsible for the difference in the plant sizes. In order to test this, Gary bought ten small plants of the same size and type. He made sure they had the same size and type of pot. He also made sure they had the same amount and 1.E.1 https://chem.libretexts.org/@go/page/402418 type of soil. Then Gary built a frame to hold a canvas roof over five of the plants while the other five were nearby but out in the sun. Gary was careful to make sure that each plant received exactly the same amount of water and plant food every day. 1. Which of the following is a reasonable statement of Gary’s hypothesis? a. Different plants have different characteristics. b. Plants that get more sunshine grow larger than plants that get less sunshine. c. Plants that grow in the shade grow larger. d. Plants that don’t receive water will die. e. Plants that receive the same amount of water and plant food will grow the same amount. 2. What scientific reason might Gary have for insisting that the container size for the all plants be the same? a. Gary wanted to determine if the size of the container would affect the plant growth. b. Gary wanted to make sure the size of the container did not affect differential plant growth in his experiment. c. Gary want to control how much plant food his plants received. d. Gary wanted his garden to look organized. e. There is no possible scientific reason for having the same size containers. 3. What scientific reason might Gary have for insisting that all plants receive the same amount of water everyday? a. Gary wanted to test the effect of shade on plant growth and therefore, he wanted to have no variables other than the amount of sunshine on the plants. b. Gary wanted to test the effect of the amount of water on plant growth. c. Gary's hypothesis was that water quality was affecting plant growth. d. Gary was conserving water. e. There is no possible scientific reason for having the same amount of water for each plant every day. 4. What was the variable being tested in Gary's experiment? a. the amount of water b. the amount of plant food c. the amount of soil d. the amount of sunshine e. the type of soil 5. Which of the following factors may be varying in Gary’s experimental setup that he did not control? a. individual plant variation b. soil temperature due to different colors of containers c. water loss due to evaporation from the soil d. the effect of insects which may attack one set of plants but not the other e. All of the above are possible factors that Gary did not control. When a mosquito sucks blood from its host, it penetrates the skin with its sharp beak and injects an anti-coagulant so the blood will not clot. It then sucks some blood and removes its beak. If the mosquito carries disease-causing microorganisms, it injects these into its host along with the anti-coagulant. It was assumed for a long time that the virus typhus was injected by the louse when sucking blood in a manner similar to the mosquito. But apparently this is not so. The infection is not in the saliva of the louse, but in the feces. The disease is thought to be spread when the louse feces come in contact with scratches or bite wounds in the host's skin. A test of this was carried out in 1922 when two workers fed infected lice on a monkey, taking great care that no louse feces came into contact with the monkey. After two weeks, the monkey had NOT become ill with typhus. The workers then injected the monkey with typhus and it became ill within a few days. Why did the workers inject the monkey with typhus near the end of the experiment? a. to prove that the lice carried the typhus virus b. to prove the monkey was similar to man c. to prove that the monkey was not immune to typhus d. to prove that mosquitoes were not carriers of typhus e. the workers were mean Eijkman fed a group of chickens exclusively on rice whose seed coat had been removed (polished rice or white rice). The chickens all developed polyneuritis (a disease of chickens) and died. He fed another group of chickens unpolished rice (rice that still had its 1.E.2 https://chem.libretexts.org/@go/page/402418 seed coat). Not a single one of them contracted polyneuritis. He then gathered the polishings from rice (the seed coats that had been removed) and fed the polishings to other chickens that were sick with polyneuritis. In a short time, the birds all recovered. Eijkman had accurately traced the cause of polyneuritis to a faulty diet. For the first time in history, a food deficiency disease had been produced and cured experimentally. Which of the following is a reasonable statement of Eijkman’s hypothesis? a. Polyneuritis is a fatal disease for chickens. b. White rice carries a virus for the disease polyneuritis. c. Unpolished rice does not carry the polyneuritis virus. d. The rice seed coat contains a nutrient that provides protection for chickens against polyneuritis. e. None of these is a reasonable statement of Eijkman's hypothesis. The three questions below relate to the following paragraphs. Scientist A noticed that in a certain forest area, the only animals inhabiting the region were giraffes. He also noticed that the only food available for the animals was on fairly tall trees and as the summer progressed, the animals ate the leaves high and higher on the trees. The scientist suggested that these animals were originally like all other animals but generations of animals stretching their necks to reach higher up the trees for food, caused the species to grow very long necks. Scientist B conducted experiments and observed that stretching muscles does NOT cause bones to grow longer nor change the DNA of animals so that longer muscles would be passed on to the next generation. Scientist B, therefore, discarded Scientist A's suggested answer as to why all the animals living in the area had long necks. Scientist B suggested instead that originally many different types of animals including giraffes had lived in the region but only the giraffes could survive when the only food was high in the trees, and so all the other species had left the area. 1. Which of the following statements is an interpretation, rather than an observation? A. The only animals living in the area were giraffes. B. The only available food was on tall trees. C. Animals which constantly stretch their necks will grow longer necks. D. A, B, and C are all interpretations. E. A, B, and C are all observations. 2. Scientist A's hypothesis was that A. the only animals living in the area were giraffes. B. the only available food was on tall trees. C. animals which constantly stretch their necks will grow longer necks. D. the animals which possess the best characteristics for living in an area, will be the predominant species. E. None of the above are reasonable statements of Scientist A's hypothesis. 3. Scientist A's hypothesis being discarded is A. evidence that the scientific method doesn’t always work. B. a result achieved without use of the scientific method. C. an example of what happened before the scientific method was invented. D. an example of the normal functioning of the scientific method. E. an unusual case. When a theory has been known for a long time, it becomes a law. a. True b. False During Pasteur's time, anthrax was a widespread and disastrous disease for livestock. Many people whose livelihood was raising livestock lost large portions of their herds to this disease. Around 1876, a horse doctor in eastern France named Louvrier, claimed to have invented a cure for anthrax. The influential men of the community supported Louvrier's claim to have cured hundreds of cows of anthrax. Pasteur went to Louvrier's hometown to evaluate the cure. The cure was explained to Pasteur as a multi-step process during which: 1) the cow was rubbed vigorously to make her as hot as possible; 2) long gashes were cut into the cows skin and turpentine was poured into the cuts; 3) an inch-thick coating of cow manure mixed with hot vinegar was plastered onto the cow and the cow was completely wrapped in a cloth. Since some cows recover from anthrax with no treatment, performing the cure on a single cow would not be conclusive, so Pasteur proposed an experiment to test Louvrier's cure. Four healthy cows were to be 1.E.3 https://chem.libretexts.org/@go/page/402418 injected with anthrax microbes, and after the cows became ill, Louvrier would pick two of the cows (A and B) and perform his cure on them while the other two cows (C and D) would be left untreated. The experiment was performed and after a few days, one of the untreated cows died and one of them got better. Of the cows treated by Louvrier's cure, one cow died and one got better. In this experiment, what was the purpose of infecting cows C and D? a. So that Louvrier would have more than two cows to choose from. b. To make sure the injection actually contained anthrax. c. To serve as experimental controls (a comparison of treated to untreated cows). d. To kill as many cows as possible. A hypothesis is a. a description of a consistent pattern in observations. b. an observation that remains constant. c. a theory that has been proven. d. a tentative explanation for a phenomenon. A number of people became ill after eating oysters in a restaurant. Which of the following statements is a hypothesis about this occurrence? a. Everyone who ate oysters got sick. b. People got sick whether the oysters they ate were raw or cooked. c. Symptoms included nausea and dizziness. d. The cook felt really bad about it. e. Bacteria in the oysters may have caused the illness. Which statement best describes the reason for using experimental controls? a. Experimental controls eliminate the need for large sample sizes. b. Experimental controls eliminate the need for statistical tests. c. Experimental controls reduce the number of measurements needed. d. Experimental controls allow comparison between groups that are different in only one independent variable. A student decides to set up an experiment to determine the relationship between the growth rate of plants and the presence of detergent in the soil. He sets up 10 seed pots. In five of the seed pots, he mixes a precise amount of detergent with the soil and the other five seed pots have no detergent in the soil. The five seed pots with detergent are placed in the sun and the five seed pots with no detergent are placed in the shade. All 10 seed pots receive the same amount of water and the same number and type of seeds. He grows the plants for two months and charts the growth every two days. What is wrong with his experiment? a. The student has too few pots. b. The student has two independent variables. c. The student has two dependent variables. d. The student has no experimental control on the soil. A scientist plants two rows of corn for experimentation. She puts fertilizer on row 1 but does not put fertilizer on row 2. Both rows receive the same amount of sun and water. She checks the growth of the corn over the course of five months. What is acting as the control in this experiment? a. Corn without fertilizer. b. Corn with fertilizer. c. Amount of water. d. Height of corn plants. If you have a control group for your experiment, which of the following is true? a. There can be more than one difference between the control group and the test group, but not more three differences, or else the experiment is invalid. b. The control group and the test group may have many differences between them. c. The control group must be identical to the test group except for one variable. d. None of these are true. 1.E.4 https://chem.libretexts.org/@go/page/402418 If the hypothesis is rejected by the experiment, then: a. the experiment may have been a success. b. the experiment was a failure. c. the experiment was poorly designed. d. the experiment didn't follow the scientific method. A well-substantiated explanation of an aspect of the natural world is a: a. theory. b. law. c. hypothesis. d. None of these. 1.5: A Beginning Chemist: How to Succeed 1.E: Exercises is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 1.E: Exercises by Henry Agnew, Marisa Alviar-Agnew is licensed CK-12. Original source: https://www.ck12.org/c/chemistry/. 1.E.5 https://chem.libretexts.org/@go/page/402418  CHAPTER OVERVIEW 2: Measurements and Problem Solving 2.1: Taking Measurements 2.2: Scientific Notation - Writing Large and Small Numbers 2.3: Significant Figures - Writing Numbers to Reflect Precision 2.4: Significant Figures in Calculations 2.5: The Basic Units of Measurement 2.6: Problem Solving and Unit Conversions 2.7: Solving Multi-step Conversion Problems 2.8: Density 2: Measurements and Problem Solving is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts. 1 2.1: TAKING MEASUREMENTS  LEARNING OBJECTIVES Express quantities properly, using a number and a unit. A coffee maker’s instructions tell you to fill the coffee pot with 4 cups of water and to use 3 scoops of coffee. When you follow these instructions, you are measuring. When you visit a doctor’s office, a nurse checks your temperature, height, weight, and perhaps blood pressure (Figure 2.1.1 ); the nurse is also measuring. Figure 2.1.1: Measuring Blood Pressure. A nurse or a doctor measuring a patient’s blood pressure is taking a measurement. (GFDL; Pia von Lützau). Chemists measure the properties of matter and express these measurements as quantities. A quantity is an amount of something and consists of a number and a unit. The number tells us how many (or how much), and the unit tells us what the scale of measurement is. For example, when a distance is reported as “5 kilometers,” we know that the quantity has been expressed in units of kilometers and that the number of kilometers is 5. If you ask a friend how far they walk from home to school, and the friend answers “12” without specifying a unit, you do not know whether your friend walks 12 kilometers, 12 miles, 12 furlongs, or 12 yards. Both a number and a unit must be included to express a quantity properly. To understand chemistry, we need a clear understanding of the units chemists work with and the rules they follow for expressing numbers. The next two sections examine the rules for expressing numbers.  EXAMPLE 2.1.1 Identify the number and the unit in each quantity. a. one dozen eggs b. 2.54 centimeters c. a box of pencils d. 88 meters per second SOLUTION a. The number is one, and the unit is a dozen eggs. b. The number is 2.54, and the unit is centimeter. c. The number 1 is implied because the quantity is only a box. The unit is box of pencils. d. The number is 88, and the unit is meters per second. Note that in this case the unit is actually a combination of two units: meters and seconds. KEY TAKE AWAY Identify a quantity properly with a number and a unit. 2.1: Taking Measurements is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 2.1: Taking Measurements by Henry Agnew, Marisa Alviar-Agnew is licensed CC BY-NC-SA 3.0. 2.1.1 https://chem.libretexts.org/@go/page/402420 2.2: Scientific Notation - Writing Large and Small Numbers  Learning Objectives Express a large number or a small number in scientific notation. Carry out arithmetical operations and express the final answer in scientific notation Chemists often work with numbers that are exceedingly large or small. For example, entering the mass in grams of a hydrogen atom into a calculator would require a display with at least 24 decimal places. A system called scientific notation avoids much of the tedium and awkwardness of manipulating numbers with large or small magnitudes. In scientific notation, these numbers are expressed in the form n N × 10 where N is greater than or equal to 1 and less than 10 (1 ≤ N < 10), and n is a positive or negative integer (100 = 1). The number 10 is called the base because it is this number that is raised to the power n. Although a base number may have values other than 10, the base number in scientific notation is always 10. A simple way to convert numbers to scientific notation is to move the decimal point as many places to the left or right as needed to give a number from 1 to 10 (N). The magnitude of n is then determined as follows: If the decimal point is moved to the left n places, n is positive. If the decimal point is moved to the right n places, n is negative. Another way to remember this is to recognize that as the number N decreases in magnitude, the exponent increases and vice versa. The application of this rule is illustrated in Example 2.2.1.  Example 2.2.1: Expressing Numbers in Scientific Notation Convert each number to scientific notation. a. 637.8 b. 0.0479 c. 7.86 d. 12,378 e. 0.00032 f. 61.06700 g. 2002.080 h. 0.01020 Solution Solutions to Example 2.2.1 Explanation Answer To convert 637.8 to a number from 1 to 10, we move the decimal point two places to the a left: 637.8 6.378 × 10 2 Because the decimal point was moved two places to the left, n = 2. To convert 0.0479 to a number from 1 to 10, we move the decimal point two places to the b right: 0.0479 4.79 × 10 −2 Because the decimal point was moved two places to the right, n = −2. This is usually expressed simply as 7.86. c 7.86 × 10 0 (Recall that 100 = 1.) 2.2.1 https://chem.libretexts.org/@go/page/402421  Explanation Answer Because the decimal point was moved four d 1.2378 × 10 4 places to the left, n = 4. Because the decimal point was moved four e 3.2 × 10 −4 places to the right, n = −4. Because the decimal point was moved one f 6.106700 × 10 1 place to the left, n = 1. Because the decimal point was moved three g 2.002080 × 10 3 places to the left, n = 3. Because the decimal point was moved two h 1.020 × 10 −2 places to the right, n = -2. Addition and Subtraction Before numbers expressed in scientific notation can be added or subtracted, they must be converted to a form in which all the exponents have the same value. The appropriate operation is then carried out on the values of N. Example 2.2.2 illustrates how to do this.  Example 2.2.2: Expressing Sums and Differences in Scientific Notation Carry out the appropriate operation and then express the answer in scientific notation. a. (1.36 × 10 ) + (4.73 × 10 ) 2 3 b. (6.923 × 10 ) − (8.756 × 10 −3 −4 ) Solution Solutions to Example 2.2.2. Explanation Answer Both exponents must have the same value, so these numbers are converted to either 2 2 2 2 (1.36 × 10 ) + (47.3 × 10 ) = (1.36 + 47.3) × 10 = 48.66 × 10 or 3 3 3 3 (0.136 × 10 ) + (4.73 × 10 ) = (0.136 + 4.73) × 10 ) = 4.87 × 10 a. 4.87 × 10 3 Choosing either alternative gives the same answer, reported to two decimal places. In converting 48.66 × 102 to scientific notation, n has become more positive by 1 because the value of N has decreased. Converting the exponents to the same value gives either −3 −3 −3 (6.923 × 10 ) − (0.8756 × 10 ) = (6.923 − 0.8756) × 10 or b (69.23 × 10 −4 ) − (8.756 × 10 −4 ) = (69.23 −6.047 8.756) ××10 10 −3−4 = 60.474 × 10 −4. In converting 60.474 × 10-4 to scientific notation, n has become more positive by 1 because the value of N has decreased. Multiplication and Division When multiplying numbers expressed in scientific notation, we multiply the values of N and add together the values of n. Conversely, when dividing, we divide N in the dividend (the number being divided) by N in the divisor (the number by which we 2.2.2 https://chem.libretexts.org/@go/page/402421 are dividing) and then subtract n in the divisor from n in the dividend. In contrast to addition and subtraction, the exponents do not have to be the same in multiplication and division. Examples of problems involving multiplication and division are shown in Example 2.2.3.  Example 2.2.3: Expressing Products and Quotients in Scientific Notation Perform the appropriate operation and express your answer in scientific notation. a. (6.022 × 10 23 )(6.42 × 10 −2 ) −24 1.67 × 10 b. −28 9.12 × 10 −34 (6.63 × 10 )(6.0 × 10) c. −2 8.52 × 10 Solution Solution to Example 2.2.3 Explanation Answer In multiplication, we add the exponents: 23 −2 [23+(−2)] 21 (6.022 × 10 )(6.42 × 10 ) = (6.022)(6.42) × 10 = 38.7 × 10 a In converting 38.7 × 10 to scientific 21 notation, n has become more positive by 1 because the value of N has decreased. 3.87 × 10 22 b In division, we subtract the exponents: −24 1.67 × 10 1.67 [−24−(−28)] 4 = × 10 = 0.183 × 10 −28 9.12 × 10 9.12 In converting 0.183 × 10 to scientific notation, n has become more negative by 1 because the value of N has increased. 4 1.83 × 10 3 c This problem has both multiplication and division: −34 (6.63 × 10 )(6.0 × 10) 39.78 [−34+1−(−2)] = × 10 −2 (8.52 × 10 ) 8.52 −31 4.7 × 10 2.2: Scientific Notation - Writing Large and Small Numbers is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew & Henry Agnew. 2.2: Scientific Notation - Writing Large and Small Numbers by Henry Agnew, Marisa Alviar-Agnew is licensed CK-12. Original source: https://www.ck12.org/c/chemistry/. 2.2.3 https://chem.libretexts.org/@go/page/402421 2.3: Significant Figures - Writing Numbers to Reflect Precision  Learning Objectives Identify the number of significant figures in a reported value. The significant figures in a measurement consist of all the certain digits in that measurement plus one uncertain or estimated digit. In the ruler illustration below, the bottom ruler gave a length with 2 significant figures, while the top ruler gave a length with 3 significant figures. In a correctly reported measurement, the final digit is significant but not certain. Insignificant digits are not reported. With either ruler, it would not be possible to report the length at 2.553 cm as there is no possible way that the thousandths digit could be estimated. The 3 is not significant and would not be reported. Figure 2.3.1 : Measurement with two different rulers. Ruler A's measurement can be rounded to 2.55, with 2 certain digits, while Ruler B's measurement of 2.5 has 1 certain digit Measurement Uncertainty Some error or uncertainty always exists in any measurement. The amount of uncertainty depends both upon the skill of the measurer and upon the quality of the measuring tool. While some balances are capable of measuring masses only to the nearest 0.1 g, other highly sensitive balances are capable of measuring to the nearest 0.001 g or even better. Many measuring tools such as rulers and graduated cylinders have small lines which need to be carefully read in order to make a measurement. Figure 2.3.1 shows two rulers making the same measurement of an object (indicated by the blue arrow). With either ruler, it is clear that the length of the object is between 2 and 3 cm. The bottom ruler contains no millimeter markings. With that ruler, the tenths digit can be estimated and the length may be reported as 2.5 cm. However, another person may judge that the measurement is 2.4 cm or perhaps 2.6 cm. While the 2 is known for certain, the value of the tenths digit is uncertain. The top ruler contains marks for tenths of a centimeter (millimeters). Now the same object may be measured as 2.55 cm. The measurer is capable of estimating the hundredths digit because he can be certain that the tenths digit is a 5. Again, another measurer may report the length to be 2.54 cm or 2.56 cm. In this case, there are two certain digits (the 2 and the 5), with the hundredths digit being uncertain. Clearly, the top ruler is a superior ruler for measuring lengths as precisely as possible.  Example 2.3.1: Reporting Measurements to the Proper Number of Significant Figures Use each diagram to report a measurement to the proper number of significant figures. a. b. 2.3.1 https://chem.libretexts.org/@go/page/402422  Ruler measuring a rectangle in units of centimeters, with the rectangle's edge between 1.2 and 1.3 cm marks Solutions Solutions to Example 2.3.1 Explanation Answer The arrow is between 4.0 and 5.0, so the measurement is at least 4.0. The arrow is between the third and fourth small tick marks, so it’s at least 0.3. We will have to estimate the last place. It looks like about a. one-third of the way across the space, so let 4.33 psi us estimate the hundredths place as 3. The symbol psi stands for “pounds per square inch” and is a unit of pressure, like air in a tire. The measurement is reported to three significant figures. The rectangle is at least 1.0 cm wide but certainly not 2.0 cm wide, so the first significant digit is 1. The rectangle’s width is past the second tick mark but not the third; if each tick mark represents 0.1, then the rectangle is at least 0.2 in the next significant b. digit. We have to estimate the next place 1.25 cm because there are no markings to guide us. It appears to be about halfway between 0.2 and 0.3, so we will estimate the next place to be a 5. Thus, the measured width of the rectangle is 1.25 cm. The measurement is reported to three significant figures.  Exercise 2.3.1 What would be the reported width of this rectangle? Answer 1.25 cm When you look at a reported measurement, it is necessary to be able to count the number of significant figures. The table below details the rules for determining the number of significant figures in a reported measurement. For the examples in the table, assume 2.3.2 https://chem.libretexts.org/@go/page/402422 that the quantities are correctly reported values of a measured quantity. Table 2.3.1 : Significant Figure Rules Rule Examples 237 has three significant figures. 1. All nonzero digits in a measurement are significant. 1.897 has four significant figures. 2. Zeros that appear between other nonzero digits (middle zeros) are 39,004 has five significant figures. always significant. 5.02 has three significant figures. 3. Zeros that appear in front of all of the nonzero digits are called 0.008 has one significant figure. leading zeros. Leading zeros are never significant. 0.000416 has three significant figures. 4. Zeros that appear after all nonzero digits are called trailing zeros. A 1400 is ambiguous. number with trailing zeros that lacks a decimal point may or may not be 1.4 × 10 has two significant figures. 3 significant. Use scientific notation to indicate the appropriate 1.40 × 10 three significant figures. 3 number of significant figures. 1.400 × 10 has four significant figures. 3 5. Trailing zeros in a number with a decimal point are significant. This 620.0 has four significant figures. is true whether the zeros occur before or after the decimal point. 19.000 has five significant figures. Exact Numbers Integers obtained either by counting objects or from definitions are exact numbers, which are considered to have infinitely many significant figures. If we have counted four objects, for example, then the number 4 has an infinite number of significant figures (i.e., it represents 4.000…). Similarly, 1 foot (ft) is defined to contain 12 inches (in), so the number 12 in the following equation has infinitely many significant figures:  Example 2.3.2 Give the number of significant figures in each. Identify the rule for each. a. 5.87 b. 0.031 c. 52.90 d. 00.2001 e. 500 f. 6 atoms Solution Solution to Example 2.3.2 Explanation Answer a All three numbers are significant (rule 1). 5.87, three significant figures The leading zeros are not significant (rule 3). b 0.031, two significant figures The 3 and the 1 are significant (rule 1). The 5, the 2 and the 9 are significant (rule 1). c 52.90, four significant figures The trailing zero is also significant (rule 5). The leading zeros are not significant (rule 3). d The 2 and the 1 are significant (rule 1) and 00.2001, four significant figures the middle zeros are also significant (rule 2). The number is ambiguous. It could have one, e 500, ambiguous two or three significant figures. The 6 is a counting number. A counting f 6, infinite number is an exact number. 2.3.3 https://chem.libretexts.org/@go/page/402422   Exercise 2.3.2 Give the number of significant figures in each. a. 36.7 m b. 0.006606 s c. 2,002 kg d. 306,490,000 people e. 3,800 g Answer a three significant figures Answer b four significant figures Answer c four significant figures Answer d infinite (exact number) Answer e Ambiguous, could be two, three or four significant figures. Accuracy and Precision Measurements may be accurate, meaning that the measured value is the same as the true value; they may be precise, meaning that multiple measurements give nearly identical values (i.e., reproducible results); they may be both accurate and precise; or they may be neither accurate nor precise. The goal of scientists is to obtain measured values that are both accurate and precise. The video below demonstrates the concepts of accuracy and precision. What's the difference between accurac… accurac… Video 2.3.1 : Difference between precision and accuracy.  Example 2.3.3 The following archery targets show marks that represent the results of four sets of measurements. 2.3.4 https://chem.libretexts.org/@go/page/402422  Which target shows a. a precise, but inaccurate set of measurements? b. a set of measurements that is both precise and accurate? c. a set of measurements that is neither precise nor accurate? Solution a. Set a is precise, but inaccurate. b. Set c is both precise and accurate. c. Set d is neither precise nor accurate. Summary Uncertainty exists in all measurements. The degree of uncertainty is affected in part by the quality of the measuring tool. Significant figures give an indication of the certainty of a measurement. Rules allow decisions to be made about how many digits to use in any given situation. 2.3: Significant Figures - Writing Numbers to Reflect Precision is shared under a CK-12 license and was authored, remixed, and/or curated by Marisa Alviar-Agnew, Henry Agnew, Sridhar Budhi, & Sridhar Budhi. 2.3: Significant Figures - Writing Numbers to Reflect Precision by Henry Agnew, Marisa Alviar-Agnew, Sridhar Budhi is licensed CK-12. Original source: https://www.ck12.org/c/chemistry/. 2.3.5 https://chem.libretexts.org/@go/page/402422 2.4: Significant Figures in Calculations  Learning Objectives Use significant figures correctly in arithmetical operations. Rounding Before dealing with the specifics of the rules for determining the significant figures in a calculated result, we need to be able to round numbers correctly. To round a number, first decide how many significant figures the number should have. Once you know that, round to that many digits, starting from the left. If the number immediately to the right of the last significant digit is less than 5, it is dropped and the value of the last significant digit remains the same. If the number immediately to the right of the last significant digit is greater than or equal to 5, the last significant digit is increased by 1. Consider the measurement 207.518 m. Right now, the measurement contains six significant figures. How would we successively round it to fewer and fewer significant figures? Follow the process as outlined in Table 2.4.1. Table 2.4.1 : Rounding examples Number of Significant Figures Rounded Value Reasoning 6 207.518 All digits are significant 5 207.52 8 rounds the 1 up to 2 4 207.5 2 is dropped 3 208 5 rounds the 7 up to 8 2 210 8 is replaced by a 0 and rounds the 0 up to 1 1 200 1 is replaced by a 0 Notice that the more rounding that is done, the less reliable the figure is. An approximate value may be sufficient for some purposes, but scientific work requires a much higher level of detail. It is important to be aware of significant figures when you are mathematically manipulating numbers. For example, dividing 125 by 307 on a calculator gives 0.4071661238… to an infinite number of digits. But do the digits in this answer have any practical meaning, especially when you are starting with numbers that have only three significant figures each? When performing mathematical operations, there are two rules for limiting the number of significant figures in an answer—one rule is for addition and subtraction, and one rule is for multiplication and division. In operations involving significant figures, the answer is reported in such a way that it reflects the reliability of the least precise operation. An answer is no more precise than the least precise number used to get the answer. Multiplication and Division For multiplication or division, the rule is to count the number of significant figures in each number being multiplied or divided and then limit the significant figures in the answer to the lowest count. An example is as follows: The final answer, limited to four significant figures, is 4,094. The first digit dropped is 1, so we do not round up. Scientific notation provides a way of communicating significant figures without ambiguity. You simply include all the significant figures in the leading number. For example, the number 450 has two significant figures and would be written in scientific notation as 4.5 × 102, whereas 450.0 has four significant figures and would be written as 4.500 × 102. In scientific notation, all significant figures are listed explicitly. 2.4.1 https://chem.libretexts.org/@go/page/402423   Example 2.4.1 Write the answer for each expression using scientific notation with the appropriate number of significant figures. a. 23.096 × 90.300 b. 125 × 9.000 Solution a Table with two columns and 1 row. The first column on the left is labeled, Explanation, and underneath in the row is an explanation. The second column is labeled, Answer, and underneath in the row is an answer. Explanation Answer The calculator answer is 2,085.5688, but we need to round it to five significant figures. Because the first digit to be dropped (in the tenths 2.0856 × 10 3 place) is greater than 5, we round up to 2,085.6. b Table with two columns and 1 row. The first column on the left is labeled, Explanation, and underneath in the row is an explanation. The second column is labeled, Answer, and underneath in the row is an answer. Explanation Answer The calculator gives 1,125 as the answer, but we limit it to three 3 1.13 × 10 significant figures. Addition and Subtraction How are significant figures handled in calculations? It depends on what type of calculation is being performed. If the calculation is an addition or a subtraction, the rule is as follows: limit the reported answer to the rightmost column that all numbers have significant figures in common. For example, if you were to add 1.2 and 4.71, we note that the first number stops its significant figures in the tenths column, while the second number stops its significant figures in the hundredths column. We therefore limit our answer to the tenths column. We drop the last digit—the 1—because it is not significant to the final answer. The dropping of positions in sums and differences brings up the topic of rounding. Although there are several conventions, in this text we will adopt the following rule: the final answer should be rounded up if the first dropped digit is 5 or greater, and rounded down if the first dropped digit is less than 5.  Example 2.4.2 a. 13.77 + 908.226 b. 1,027 + 611 + 363.06 2.4.2 https://chem.libretexts.org/@go/page/402423  Solution a Table with two columns and 1 row. The first column on the left is labeled, Explanation, and underneath in the row is an explanation. The second column is labeled, Answer, and underneath in the row is an answer. Explanation Answer The calculator answer is 921.996, but because 13.77 has its farthest- right significant figure in the hundredths place, we need to round the final answer to the hundredths position. Because the first digit to be 922.00 = 9.2200 × 10 2 dropped (in the thousandths place) is greater than 5, we round up to 922.00 b Table with two columns and 1 row. The first column on the left is labeled, Explanation, and underneath in the row is an explanation. The second column is labeled, Answer, and underneath in the row is an answer. Explanation Answer The calculator gives 2,001.06 as the answer, but because 611 and 1027 has its farthest-right significant figure in the ones place, the 2, 001.06 = 2.001 × 10 3 final answer must be limited to the ones position.  Exercise 2.4.2 Write the answer for each expression using scientific notation with the appropriate number of significant figures. a. 217 ÷ 903 b. 13.77 + 908.226 + 515 c. 255.0 − 99 d. 0.00666 × 321 Answer a: −1 0.240 = 2.40 × 10 Answer b: 3 1, 437 = 1.437 × 10 Answer c: 2 156 = 1.56 × 10 Answer d: 0 2.14 = 2.14 × 10 Remember that calculators do not understand significant figures. You are the one who must apply the rules of significant figures to a result from your calculator. Calculations Involving Multiplication/Division a