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Introduction to Chemistry: LAB1.1 Expressing Numbers

Introduction to Chemistry
LAB1.1 Expressing Numbers
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table of contents
  1. Cover
  2. Title Page
  3. Copyright
  4. Table Of Contents
  5. Introduction
  6. Preface
  7. Acknowledgements
  8. About the Authors
  9. Chapter 1. Chemistry: An Experimental Science
    1. 1.0 Introduction
    2. 1.1 Chemistry in Context
    3. 1.2 Phases and Classification of Matter
    4. 1.3 Physical and Chemical Properties
  10. Chapter 2. Atoms, Molecules, and Ions
    1. 2.0 Introduction
    2. 2.1 Atomic Theory
    3. 2.2 Beyond Dalton’s Atomic Theory
    4. 2.3 Atomic Structure and Symbols
    5. 2.4 Chemical Formulas
    6. 2.5 The Periodic Table
  11. Chapter 3. Ions, Bonding and Compound Formation
    1. 3.0 Introduction
    2. 3.1 Ionic and Molecular Compounds
    3. 3.2 Nomenclature of Simple Ionic and Molecular Compounds
  12. Chapter 4. Chemical Reactions and Equations
    1. 4.0 Introduction
    2. 4.1 Writing and Balancing Chemical Equations
    3. 4.2 Precipitation Reactions
    4. 4.3 Acid-Base Reactions
    5. 4.4 Oxidation-Reduction Reactions
  13. Chapter 5. Bridging the Macroscopic and Microscopic Realms
    1. 5.0 Introduction
    2. 5.1 Formula Mass
    3. 5.2 The Mole
    4. 5.3 Reaction Stoichiometry
    5. 5.4 Limiting Reactant and Reaction Yields
  14. Chapter 6. Solutions
    1. 6.0 Introduction
    2. 6.1 Solution Concentration and Molarity
    3. 6.2 Other Concentration Units
  15. Chapter 7. Chemical Bonding and Lewis Structures
    1. 7.0 Introduction
    2. 7.1 Covalent Bonding
    3. 7.2 Lewis Dot Structures
    4. 7.3 Lewis Structures and Covalent Compounds
  16. Additional Reading: Electronic Structure of Atoms
    1. 8.0 Introduction
    2. 8.1 Electromagnetic Energy
    3. 8.2 Quantization of the Energy of Electrons
    4. 8.3 Development of Quantum Theory
    5. 8.4 Electronic Structure of Atoms
    6. 8.5 Periodic Trends
  17. Chapter LAB1. Making Measurements
    1. Introduction
    2. LAB1.1 Expressing Numbers
    3. LAB1.2 Measurements and Units
    4. LAB1.3 Measurement Uncertainty, Accuracy, and Precision
    5. LAB1.4 Mathematical Treatment of Measurement Results – Unit Conversions
    6. LAB1.5 Density – A Derived Unit and Conversion Factor
  18. Appendix
  19. Appendix A: The Periodic Table
  20. Appendix B: Essential Mathematics
  21. Appendix C: Units and Conversion Factors
  22. Appendix D: Fundamental Physical Constants

38

LAB1.1 Expressing Numbers

Learning Objective

By the end of this section, you will be able to:

  • Use scientific notation properly when you express and work with numbers.

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:

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:

10−1= 0.1 = 1/10
10−2= 0.01 = 1/100
10−3= 0.001 = 1/1,000
10−4= 0.0001 = 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 × 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 1

Express these numbers in scientific notation.

a) 306,000          b) 0.00884          c) 2,760,000          d) 0.000000559

Solution

a) The number 306,000 is 3.06 times 100,000, or 3.06 times 105. In scientific notation, the number is 3.06 × 105.

b) The number 0.00884 is 8.84 times 1/1,000, which is 8.84 times 10−3. In scientific notation, the number is 8.84 × 10−3.

c) 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.

d) The number 0.000000559 is 5.59 times 1/10,000,000, which is 5.59 times 10−7. In scientific notation, the number is written as 5.59 × 10−7.

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.

Example 2

Represent the following numbers in scientific notation.
a)  12,500               b)  1470                c)  0.0024

Solution

a) Move the decimal until you have a number between 1 and 10:
Because the decimal was moved 4 places to the left, the power of ten is 4.
The answer is 25 x 104

b) The decimal was moved 3 places to the left, the answer is  47 x 103

c) The decimal was moved 3 places to the right, the answer is  4 x 10-3

Example 3

Represent the following scientific notation numbers in normal notation
a)  3.06 x 10-2              b) 2.47 x 106

Solution

a) You now have to move the decimal in the opposite direction, thus a negative power of 10

requires a movement to the right (2 spaces):  3.06  →  0.0306

b) The positive exponent requires a movement 6 spaces to the left, adding zeros as required.
2.47 →  2,470,000

Key Concepts and Summary

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.

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.  Source: “Casio”Asim Bijarani is licensed under Creative Commons Attribution 2.0 Generic.

Review-Refresh, Extend

Review-Refresh

1.  Express these numbers in scientific notation.

a)  −890,000          b)  602,000,000,000          c)  0.0000004099          d)  0.000000000000011

2.  Express these numbers in standard notation.

a)  7.11 × 10−2          b)  9.18 × 102          c)  3.09 × 10−10

3. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.

a)  72.44 × 103           b)  9,943 × 10−5           c)  588,399 × 102

4.  Write these numbers in scientific notation by counting the number of places the decimal point is moved.

a)  123,456.78          b)  98,490          c)  0.000000445

5.  Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.

a)  456 × (7.4 × 108) = ?               b)  (3.02 × 105) ÷ (9.04 × 1015) = ?                c)  0.0044 × 0.000833 = ?

6.  Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.

a)  98,000 × 23,000 = ?              b)  98,000 ÷ 23,000 = ?              c)  (4.6 × 10−5) × (2.09 × 103) = ?

Extend

  1. The abbreviations used to describe computing memory are probably somewhat familiar to you: KB, MB, GB, TB. What are the translations of those abbreviations, and what is their relationship to the numbers in standard notation? In other words, what number describes 5.12 KB, or other types of numbers using these abbreviations?

Answers

1. a)  −8.9 × 105          b)  6.02 × 1011          c)  4.099 × 10-7          d)  1.1 × 10-14

2. a)  0.0711          b)  918          c)  0.000000000309

3. a)  7.244 × 104          b)   9.943 × 10−2          c)  5.88399 × 107

4. a)  1.2345678 × 105          b)  9.849 × 104          c)  4.45 × 10−7

5. a) 3.3744 × 1011          b) 3.3407 × 10−11          c) 3.665 × 10−6

6. a) 2.254 × 109               b)  4.2609 × 100             c)  9.614 × 10-2

Annotate

Next Chapter
LAB1.2 Measurements and Units
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Chemistry
Copyright © 2020 by Carol Higginbotham. Introduction to Chemistry by Carol Higginbotham is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
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