1. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 1 Exponential Astonishment 8

2. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 2 Unit 8D Logarithmic Scales: Earthquakes, Sounds, and Acids

3. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 3 SOURCE: U.S. Geological Survey The map shows the distribution of earthquakes around the world. Each dot represents an earthquake. Distribution of Earthquakes

4. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 4 The magnitude scale for earthquakes is defined so that each magnitude represents about 32 times as much energy as the prior magnitude. The magnitude, M, is related to the released energy, E, by the following equivalent formulas: log 10 E = 4.4 + 1.5M or E =(2.5 x 10 4 ) x 10 1.5M The energy is measured in joules; magnitudes have no units. The Magnitude Scale for Earthquakes

5. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 5 Using the formula for earthquake magnitudes, calculate precisely how much more energy is released for each 1 magnitude on the earthquake scale. Also find the energy change for a 0.5 change in magnitude. Solution We look at the formula that gives the energy: E = (2.5 × 104) × 10 1.5M Example

6. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 6 The first term, 2.5 × 10 4, is a constant number that is the same no matter what value we use for M. The magnitude appears only in the second term, 10 1.5M. Each time we raise the magnitude by 1, such as from 5 to 6 or from 7 to 8, the total energy E increases by a factor of 10 1.5. Each successive magnitude represents 10 1.5 ≈ 31.623 times as much energy as the prior magnitude. Each change of 1 magnitude corresponds to approx. 32 times as much energy. A change of 0.5 magnitude corresponds to a factor of 10 1.5 × 0.5 = 10 0.75 ≈ 5.6 in energy. Example (cont)

7. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 7 The 1989 San Francisco earthquake, in which 90 people were killed, had magnitude 7.1. Calculate the energy released, in joules. Compare the energy of this earthquake to that of the 2003 earthquake that destroyed the ancient city of Bam, Iran, which had magnitude 6.3 and killed an estimated 50,000 people. Solution The energy released by the San Francisco earthquake was E = (2.5 × 10 4 ) × 10 1.5M = (2.5 × 10 4 ) × 10 1.5 × 7.1 ≈ 1.1 × 10 15 joules Example

8. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 8 The San Francisco earthquake was 7.1 – 6.3 = 0.8 magnitude greater than the Iran earthquake. It therefore released 10 1.5 × 0.8 = 10 1.2 ≈ 16 times as much energy. Nevertheless, the Iran earthquake killed many more people, because more buildings collapsed. Example (cont)

9. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 9 Measuring Sound The decibel scale is used to compare the loudness of sounds. The loudness of a sound in decibels is defined by the following equivalent formulas:

10. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 10 Typical Sounds in Decibels

11. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 11 Suppose a sound is 100 times as intense as the softest audible sound. What is its loudness, in decibels? Solution We are looking for the loudness in decibels, so we use the first form of the decibel scale formula: Example

12. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 12 The ratio in parentheses is 100, because we are given that the sound is 100 times as intense as the softest audible sound. We find loudness in dB = 10 log 10 100 = 10 × 2 = 20 dB A sound that is 100 times as intense as the softest possible sound has a loudness of 20 dB, which is equivalent to a whisper. Example (cont)

13. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 13 How does the intensity of a 57-dB sound compare to that of a 23-dB sound? Solution We can compare the loudness of two sounds by working with the second form of the decibel scale formula. You should confirm for yourself that by dividing the intensity of Sound 1 by the intensity of Sound 2, we find Example

14. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 14 Substituting 57 dB for Sound 1 and 23 dB for Sound 2, we have A sound of 57 dB is about 2500 times as intense as a sound of 23 dB. Example (cont)

15. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 15 The intensity of sound decreases with the square of the distance from the source, meaning that the intensity is proportional to 1/d 2. Therefore, sound follows and inverse square law with distance. How many times greater is the intensity of sound from a concert speaker at a distance of 10 meters than the intensity at a distance of 80 meters? The sound intensity at 10 meters is 64 times the sound intensity at 80 meters. The Inverse Square Law for Sound

16. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 16 How far should you be from a jet to avoid a strong risk of damage to your ear? Solution Table 8.5 shows that the sound from a jet at a distance of 30 meters is 140 dB, and 120 dB is the level of sound that poses a strong risk of ear damage. The ratio of the intensity of these two sounds is Example

17. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 17 The sound of the jet at 30 meters is 100 times as intense as a sound that presents a strong risk of ear damage. To prevent ear damage, you must therefore be far enough from the jet to weaken this sound intensity by at least a factor of 100. Because sound intensity follows an inverse square law with distance, moving 10 times as far away weakens the intensity by a factor of 10 2 = 100. You should therefore be more than 10 × 30 m = 300 meters from the jet to be safe. Example (cont)

18. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 18 pH Scale The pH scale is used to classify substances as neutral, acidic, or basic. The pH scale is defined by the equivalent formula pH =  log 10 [H + ] or [H + ] = 10  pH where [H + ] is the hydrogen ion concentration in moles per liter. Pure water is neutral and has a pH of 7. Acids have a pH lower than 7 and bases have a pH higher than 7.

19. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 19 Typical pH Values

20. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit D, Slide 20 What is the pH of a solution with a hydrogen ion concentration of 10  12 mole per liter? Is it an acid or a base? Solution Using the first version of the pH formula with a hydrogen ion concentration of [H + ] = 10  12 mole per liter, we find pH = –log 10 [H + ] = –log 10 10 -12 = –(–12) = 12 Because this pH is well above 7, the solution is a strong base. Example