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Exponential Astonishment. 8. Unit 8D. Logarithmic Scales: Earthquakes, Sounds, and Acids. Distribution of Earthquakes. The map shows the distribution of earthquakes around the world. Each dot represents an earthquake. SOURCE: U.S. Geological Survey. The Magnitude Scale for Earthquakes.
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Unit 8D Logarithmic Scales: Earthquakes, Sounds, and Acids
Distribution of Earthquakes The map shows the distribution of earthquakes around the world. Each dot represents an earthquake. SOURCE: U.S. Geological Survey
The Magnitude Scale for Earthquakes • The magnitude scalefor 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: log10E = 4.4 + 1.5M or E =(2.5 x 104) x 101.5M • The energy is measured in joules; magnitudes have no units.
Example 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) × 101.5M
Example (cont) The first term, 2.5 × 104, is a constant number that is the same no matter what value we use for M. The magnitude appears only in the second term, 101.5M. 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 101.5. Each successive magnitude represents 101.5 ≈ 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 101.5×0.5 = 100.75 ≈ 5.6 in energy.
Example 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 × 104) × 101.5M= (2.5 × 104) × 101.5× 7.1 ≈1.1 × 1015 joules
Example (cont) The San Francisco earthquake was 7.1 – 6.3 = 0.8 magnitude greater than the Iran earthquake. It therefore released 101.5× 0.8 = 101.2 ≈ 16 times as much energy. Nevertheless, the Iran earthquake killed many more people, because more buildings collapsed.
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:
Example 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 (cont) 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 log10100 = 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 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 (cont) 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.
The Inverse Square Law for Sound The intensity of sound decreases with the square of the distance from the source, meaning that the intensity is proportional to 1/d2. 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.
Example 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 (cont) 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 102 = 100. You should therefore be more than 10 × 30 m = 300 meters from the jet to be safe.
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 = log10[H+] or [H+] = 10pH 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.
Example What is the pH of a solution with a hydrogen ion concentration of 1012 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 = –log10[H+] = –log1010-12 = –(–12) = 12 Because this pH is well above 7, the solution is a strong base.
