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Introductory Chemistry - 1st Canadian Edition: Expressing Numbers

Introductory Chemistry - 1st Canadian Edition
Expressing Numbers
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table of contents
  1. Cover
  2. Title Page
  3. Copyright
  4. Table Of Contents
  5. Acknowledgments
  6. Dedication
  7. About BCcampus Open Education
  8. Chapter 1. What is Chemistry
    1. Some Basic Definitions
    2. Chemistry as a Science
  9. Chapter 2. Measurements
    1. Expressing Numbers
    2. Significant Figures
    3. Converting Units
    4. Other Units: Temperature and Density
    5. Expressing Units
    6. End-of-Chapter Material
  10. Chapter 3. Atoms, Molecules, and Ions
    1. Acids
    2. Ions and Ionic Compounds
    3. Masses of Atoms and Molecules
    4. Molecules and Chemical Nomenclature
    5. Atomic Theory
    6. End-of-Chapter Material
  11. Chapter 4. Chemical Reactions and Equations
    1. The Chemical Equation
    2. Types of Chemical Reactions: Single- and Double-Displacement Reactions
    3. Ionic Equations: A Closer Look
    4. Composition, Decomposition, and Combustion Reactions
    5. Oxidation-Reduction Reactions
    6. Neutralization Reactions
    7. End-of-Chapter Material
  12. Chapter 5. Stoichiometry and the Mole
    1. Stoichiometry
    2. The Mole
    3. Mole-Mass and Mass-Mass Calculations
    4. Limiting Reagents
    5. The Mole in Chemical Reactions
    6. Yields
    7. End-of-Chapter Material
  13. Chapter 6. Gases
    1. Pressure
    2. Gas Laws
    3. Other Gas Laws
    4. The Ideal Gas Law and Some Applications
    5. Gas Mixtures
    6. Kinetic Molecular Theory of Gases
    7. Molecular Effusion and Diffusion
    8. Real Gases
    9. End-of-Chapter Material
  14. Chapter 7. Energy and Chemistry
    1. Formation Reactions
    2. Energy
    3. Stoichiometry Calculations Using Enthalpy
    4. Enthalpy and Chemical Reactions
    5. Work and Heat
    6. Hess’s Law
    7. End-of-Chapter Material
  15. Chapter 8. Electronic Structure
    1. Light
    2. Quantum Numbers for Electrons
    3. Organization of Electrons in Atoms
    4. Electronic Structure and the Periodic Table
    5. Periodic Trends
    6. End-of-Chapter Material
  16. Chapter 9. Chemical Bonds
    1. Lewis Electron Dot Diagrams
    2. Electron Transfer: Ionic Bonds
    3. Covalent Bonds
    4. Other Aspects of Covalent Bonds
    5. Violations of the Octet Rule
    6. Molecular Shapes and Polarity
    7. Valence Bond Theory and Hybrid Orbitals
    8. Molecular Orbitals
    9. End-of-Chapter Material
  17. Chapter 10. Solids and Liquids
    1. Properties of Liquids
    2. Solids
    3. Phase Transitions: Melting, Boiling, and Subliming
    4. Intermolecular Forces
    5. End-of-Chapter Material
  18. Chapter 11. Solutions
    1. Colligative Properties of Solutions
    2. Concentrations as Conversion Factors
    3. Quantitative Units of Concentration
    4. Colligative Properties of Ionic Solutes
    5. Some Definitions
    6. Dilutions and Concentrations
    7. End-of-Chapter Material
  19. Chapter 12. Acids and Bases
    1. Acid-Base Titrations
    2. Strong and Weak Acids and Bases and Their Salts
    3. Brønsted-Lowry Acids and Bases
    4. Arrhenius Acids and Bases
    5. Autoionization of Water
    6. Buffers
    7. The pH Scale
    8. End-of-Chapter Material
  20. Chapter 13. Chemical Equilibrium
    1. Chemical Equilibrium
    2. The Equilibrium Constant
    3. Shifting Equilibria: Le Chatelier’s Principle
    4. Calculating Equilibrium Constant Values
    5. Some Special Types of Equilibria
    6. End-of-Chapter Material
  21. Chapter 14. Oxidation and Reduction
    1. Oxidation-Reduction Reactions
    2. Balancing Redox Reactions
    3. Applications of Redox Reactions: Voltaic Cells
    4. Electrolysis
    5. End-of-Chapter Material
  22. Chapter 15. Nuclear Chemistry
    1. Units of Radioactivity
    2. Uses of Radioactive Isotopes
    3. Half-Life
    4. Radioactivity
    5. Nuclear Energy
    6. End-of-Chapter Material
  23. Chapter 16. Organic Chemistry
    1. Hydrocarbons
    2. Branched Hydrocarbons
    3. Alkyl Halides and Alcohols
    4. Other Oxygen-Containing Functional Groups
    5. Other Functional Groups
    6. Polymers
    7. End-of-Chapter Material
  24. Chapter 17. Kinetics
    1. Factors that Affect the Rate of Reactions
    2. Reaction Rates
    3. Rate Laws
    4. Concentration–Time Relationships: Integrated Rate Laws
    5. Activation Energy and the Arrhenius Equation
    6. Reaction Mechanisms
    7. Catalysis
    8. End-of-Chapter Material
  25. Chapter 18. Chemical Thermodynamics
    1. Spontaneous Change
    2. Entropy and the Second Law of Thermodynamics
    3. Measuring Entropy and Entropy Changes
    4. Gibbs Free Energy
    5. Spontaneity: Free Energy and Temperature
    6. Free Energy under Nonstandard Conditions
    7. End-of-Chapter Material
  26. Appendix A: Periodic Table of the Elements
  27. Appendix B: Selected Acid Dissociation Constants at 25°C
  28. Appendix C: Solubility Constants for Compounds at 25°C
  29. Appendix D: Standard Thermodynamic Quantities for Chemical Substances at 25°C
  30. Appendix E: Standard Reduction Potentials by Value
  31. Glossary
  32. About the Authors
  33. Versioning History

Expressing Numbers

Learning Objectives

  1. Learn to express numbers properly.

Quantities have two parts: the number and the unit. The number tells “how many.” It is important to be able to express numbers properly so that the quantities can be communicated properly.

Standard notation is the straightforward expression of a number. Numbers such as 17, 101.5, and 0.00446 are expressed in standard notation. For relatively small numbers, standard notation is fine. However, for very large numbers, such as 306,000,000, or for very small numbers, such as 0.000000419, standard notation can be cumbersome because of the number of zeros needed to place nonzero numbers in the proper position.

Scientific notation is an expression of a number using powers of 10. Powers of 10 are used to express numbers that have many zeros:

Table 2.1 Powers of 10
100= 1
101= 10
102= 100 = 10 × 10
103= 1,000 = 10 × 10 × 10
104= 10,000 = 10 × 10 × 10 × 10

and so forth. The raised number to the right of the 10 indicating the number of factors of 10 in the original number is the exponent. (Scientific notation is sometimes called exponential notation.) The exponent’s value is equal to the number of zeros in the number expressed in standard notation.

Small numbers can also be expressed in scientific notation but with negative exponents:

Table 2.2 Powers of Negative 10
10−1= 0.1 = \dfrac{1}{10}
10−2= 0.01 = \dfrac{1}{100}
10−3= 0.001 = \dfrac{1}{1,000}
10−4= 0.0001 = \dfrac{1}{10,000}

and so forth. Again, the value of the exponent is equal to the number of zeros in the denominator of the associated fraction. A negative exponent implies a decimal number less than one.

A number is expressed in scientific notation by writing the first nonzero digit, then a decimal point, and then the rest of the digits. The part of a number in scientific notation that is multiplied by a power of 10 is called the coefficient. Then determine the power of 10 needed to make that number into the original number and multiply the written number by the proper power of 10. For example, to write 79,345 in scientific notation,

79,345 = 7.9345 × 10,000 = 7.9345 × 104

Thus, the number in scientific notation is 7.9345 × 104. For small numbers, the same process is used, but the exponent for the power of 10 is negative:

0.000411 = 4.11\times \dfrac{1}{10,000} = 4.11 × 10^{-4}

Typically, the extra zero digits at the end or the beginning of a number are not included.

Example 2.1

Problems

Express these numbers in scientific notation.

  1. 306,000
  2. 0.00884
  3. 2,760,000
  4. 0.000000559

Solutions

  1. The number 306,000 is 3.06 times 100,000, or 3.06 times 105. In scientific notation, the number is 3.06 × 105.
  2. The number 0.00884 is 8.84 times \frac{1}{1,000}, which is 8.84 times 10−3. In scientific notation, the number is 8.84 × 10−3.
  3. The number 2,760,000 is 2.76 times 1,000,000, which is the same as 2.76 times 106. In scientific notation, the number is written as 2.76 × 106. Note that we omit the zeros at the end of the original number.
  4. The number 0.000000559 is 5.59 times \frac{1}{10,000,000}, which is 5.59 times 10−7. In scientific notation, the number is written as 5.59 × 10−7.

Test Yourself

Express these numbers in scientific notation.

  1. 23,070
  2. 0.0009706

Answers

  1. 2.307 × 104
  2. 9.706 × 10−4

Another way to determine the power of 10 in scientific notation is to count the number of places you need to move the decimal point to get a numerical value between 1 and 10. The number of places equals the power of 10. This number is positive if you move the decimal point to the right and negative if you move the decimal point to the left.

Many quantities in chemistry are expressed in scientific notation. When performing calculations, you may have to enter a number in scientific notation into a calculator. Be sure you know how to correctly enter a number in scientific notation into your calculator. Different models of calculators require different actions for properly entering scientific notation. If in doubt, consult your instructor immediately.

Key Takeaways

  • Standard notation expresses a number normally.
  • Scientific notation expresses a number as a coefficient times a power of 10.
  • The power of 10 is positive for numbers greater than 1 and negative for numbers between 0 and 1.
Calculator says 3.84951 times 10 to the power of 18.
This calculator shows only the coefficient and the power of 10 to represent the number in scientific notation. Thus, the number being displayed is 3.84951 × 1018, or 3,849,510,000,000,000,000.

Exercises

Questions

  1. Express these numbers in scientific notation.
    1. 56.9
    2. 563,100
    3. 0.0804
    4. 0.00000667
  2. Express these numbers in scientific notation.
    1. −890,000
    2. 602,000,000,000
    3. 0.0000004099
    4. 0.000000000000011
  3. Express these numbers in scientific notation.
    1. 0.00656
    2. 65,600
    3. 4,567,000
    4. 0.000005507
  4. Express these numbers in scientific notation.
    1. 65
    2. −321.09
    3. 0.000077099
    4. 0.000000000218
  5. Express these numbers in standard notation.
    1. 1.381 × 105
    2. 5.22 × 10−7
    3. 9.998 × 104
  6. Express these numbers in standard notation.
    1. 7.11 × 10−2
    2. 9.18 × 102
    3. 3.09 × 10−10
  7. Express these numbers in standard notation.
    1. 8.09 × 100
    2. 3.088 × 10−5
    3. −4.239 × 102
  8. Express these numbers in standard notation.
    1. 2.87 × 10−8
    2. 1.78 × 1011
    3. 1.381 × 10−23
  9. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
    1. 72.44 × 103
    2. 9,943 × 10−5
    3. 588,399 × 102
  10. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
    1. 0.000077 × 10−7
    2. 0.000111 × 108
    3. 602,000 × 1018
  11. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
    1. 345.1 × 102
    2. 0.234 × 10−3
    3. 1,800 × 10−2
  12. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
    1. 8,099 × 10−8
    2. 34.5 × 100
    3. 0.000332 × 104
  13. Write these numbers in scientific notation by counting the number of places the decimal point is moved.
    1. 123,456.78
    2. 98,490
    3. 0.000000445
  14. Write these numbers in scientific notation by counting the number of places the decimal point is moved.
    1. 0.000552
    2. 1,987
    3. 0.00000000887
  15. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
    1. 456 × (7.4 × 108) = ?
    2. (3.02 × 105) ÷ (9.04 × 1015) = ?
    3. 0.0044 × 0.000833 = ?
  16. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
    1. 98,000 × 23,000 = ?
    2. 98,000 ÷ 23,000 = ?
    3. (4.6 × 10−5) × (2.09 × 103) = ?
  17. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
    1. 45 × 132 ÷ 882 = ?
    2. [(6.37 × 104) × (8.44 × 10−4)] ÷ (3.2209 × 1015) = ?
  18. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
    1. (9.09 × 108) ÷ [(6.33 × 109) × (4.066 × 10−7)] = ?
    2. 9,345 × 34.866 ÷ 0.00665 = ?

Answers

    1. 5.69 × 101
    2. 5.631 × 105
    3. 8.04 × 10−2
    4. 6.67 × 10−6
    1. 6.56 × 10−3
    2. 6.56 × 104
    3. 4.567 × 106
    4. 5.507 × 10−6
    1. 138,100
    2. 0.000000522
    3. 99,980
    1. 8.09
    2. 0.00003088
    3. −423.9
    1. 7.244 × 104
    2. 9.943 × 10−2
    3. 5.88399 × 107
    1. 3.451 × 104
    2. 2.34 × 10−4
    3. 1.8 × 101
    1. 1.2345678 × 105
    2. 9.849 × 104
    3. 4.45 × 10−7
    1. 3.3744 × 1011
    2. 3.3407 × 10−11
    3. 3.665 × 10−6
    1. 6.7346 × 100
    2. 1.6691 × 10−14

Media Attributions

Key Takeaways

  • “Casio” by Asim Bijarani © CC BY (Attribution)

Annotate

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Significant Figures
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Chemistry

Copyright © 2014

                                by Jessie A. Key

            Introductory Chemistry - 1st Canadian Edition by Jessie A. Key is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
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