section 8d logarithm scales: earthquakes, sounds, and acids pages 546-558

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Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

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Page 1: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Section 8DLogarithm Scales: Earthquakes, Sounds, and Acids

Pages 546-558

Page 2: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Measurement Scales

Earthquake strength is described in magnitude.Loudness of sounds is described in decibels.Acidity of solutions is described by pH.

Each of these measurement scales involves exponential growth.

e.g. An earthquake of magnitude 8 is 32 times more powerful than an earthquake of magnitude 7.

e.g. A liquid with pH 5 is ten times more acidic than one with pH 6.

Page 3: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

The magnitude scale for earthquakes is defined so that each magnitude represents about 32 times as much energy as the prior magnitude.

Earthquake Magnitude Scale

Given the magnitude M we compute the releasedenergy E using the following formula:

E = (2.5 x 104) x 101.5M

Energy is measured in joules.Magnitudes have no units.

NOTE: exponential growth

Page 4: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

ex1/548 Calculate precisely how much more energy is released for each 1 magnitude on the earthquake scale (about 32 times more).

Magnitude 1: E = (2.5 x 104) x 101.5(1)

Magnitude 2: E = (2.5 x 104) x 101.5(2)

Magnitude 3: E = (2.5 x 104) x 101.5(3)

For each 1 magnitude, 101.5 = 31.623 times more energy is released.

Page 5: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

ex2/548 The 1989 San Francisco earthquake, in which 90 people were killed, had magnitude 7.1. 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.

SF in 1989 with M=7.1: E = (2.5 x 104) x 101.5(7.1)

= 1.11671E15 joules

Iran in 2003 with M=6.3: E = (2.5 x 104) x 101.5(6.3) = 7.04596E13 joules

Since 1.11671E15/7.04596E13 = 15.8489464, we say that:

the SF quake was about 16 times more powerful than the Iran quake.

Page 6: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Earthquake Magnitude Scale

Given the magnitude M we compute the releasedenergy E (joules) using the following formula:

E = (2.5 x 104) x 101.5M

Given the released energy E, how do we compute the the magnitude M?

Use common logarithms

Page 7: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Common Logarithms (page 531)

log10(x) is the power to which 10 must be raised to obtain x.

log10(x) recognizes x as a power of 10

log10(x) = y if and only if 10y = x

log10(1000) = 3 since 103 = 1000.

log10(10,000,000) = 7 since 107 = 10,000,000.

log10(1) = 0 since 100 = 1.

log10(0.1) = -1 since 10-1 = 0.1.

log10(30) = 1.4777 since 101.4777 = 30. [calculator]

Page 8: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Common Logarithms (page 531)

Page 9: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Practice with Logarithms (page 533)

23/533 100.928 is between 10 and 100.

25/533 10-5.2 is between 100,000 and 1,000,000.

27/533 is between 0 and 1.

29/533 log10(1,600,000) is between 6 and 7.

31/533 log10(0.25) is between 0 and 1.

10log ( )

Page 10: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Properties of Logarithms (page 531)

log10(10x) = x

log10(xy) = log10(x) + log10(y)

log10(ab) = b x log10(a)

10log ( )10 x x

Practice (page 533)

log10(x) is the power to which 10 must be raised to obtain x.

log10(x) recognizes x as a power of 10

log10(x) = y if and only if 10y = x

Page 11: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Earthquake Magnitude Scale

Given the magnitude M we compute the releasedenergy E (joules) using the following formula:

E = (2.5 x 104) x 101.5M

Given the released energy E, how do we compute the the magnitude M?

log10E = log10[(2.5 x 104) x 101.5M]

log10E = log10(2.5 x 104) + log10(101.5M)

log10E = 4.4 + 1.5M

Page 12: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Earthquake Magnitude Scale

Given the magnitude M we compute the releasedenergy E (joules) using the following formula:

E = (2.5 x 104) x 101.5M

Given the released energy E (joules), we compute the the magnitude M using the following formula:

log10E = 4.4 + 1.5M

More Practice 24*/554: E = 8 x 108 joules

exponential

logarithmic

Page 13: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Typical Sounds in Decibels

Decibels Times Louder than Softest

Audible Sound

Example

140 1014 jet at 30 meters

120 1012 strong risk of damage to ear

100 1010 siren at 30 meters

90 109 threshold of pain for ear

80 108 busy street traffic

60 106 ordinary conversation

40 104 background noise

20 102 whisper

10 10 rustle of leaves

0 1 threshold of human hearing

-10 0.1 inaudible sound

decibels increase by 10s and intensity is multiplied by 10s.

Page 14: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Decibel Scale for Sound

The loudness of a sound in decibels is defined by the following equivalent formulas:

10loudness in dB = 10 log ( )intensity of thesoundintensityof softest audible sound

1010loudness in dBintensity of sound

intensity of softest audible sound

KEY: How does sound compare to softest audible sound?

Page 15: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

More Practice

25/554 How many times as loud as the softest audible sound is the sound of ordinary conversation? Verify the decibel calculation on page 549.

27/554 What is the loudness, in decibels, of a sound 20 million times as loud as the softest audible sound?

29*/554 How much louder (more intense) is a 25-dB sound than a 10-dB sound?

Page 16: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

pH Scale for Acidity

The pH is used by chemists to classify substances as neutral, acidic, or basic/alkaline.

Pure Water is neutral and has a pH of 7.

Acids have a pH lower than 7.

Bases have a pH higher than 7

Solution pH Solution pH

Pure water 7 Drinking water 6.5-8.2

Stomach acid 2-3 Baking soda 8.4

Vinegar 3 Household ammonia

10

Lemon Juice 2 Drain opener 10-12

Page 17: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

The pH Scale

The pH Scale is defined by the following equivalent formulas:

pH = -log10[H+] or [H+] = 10-pH

where [H+] is the hydrogen ion concentration in moles per liter.

Page 18: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Practice

37/555 What is the hydrogen ion concentration of a solution with pH 7.5?

39/555 What is the pH of a solution with a hydrogen ion concentration of 0.01 mole per liter? Is this solution an acid or base?

Page 19: Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids Pages 546-558

Homework

Pages 554 - 555# 22,24, 28, 38,40