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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

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### Section 8DLogarithm Scales: Earthquakes, Sounds, and Acids

Pages 546-558

Earthquake strength isdescribed in magnitude.

Loudness of sounds isdescribed in decibels.

Acidity of solutions isdescribed 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.

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

Given the magnitude M we compute the released

energy E using the following formula:

E = (2.5 x 104) x 101.5M

Energyis measured injoules.

Magnitudeshaveno units.

NOTE: exponential growth

ex1/548Calculate 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.

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.

Given the magnitude M we compute the released

energy 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

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) = 3since103 = 1000.

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

log10(1) = 0since 100 = 1.

log10(0.1) = -1since10-1 = 0.1.

log10(30) = 1.4777since101.4777 = 30. [calculator]

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.

Properties of 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(10x) = x

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

log10(ab) = b x log10(a)

Practice (page 533)

Given the magnitude M we compute the released

energy 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

Given the magnitude M we compute the released

energy E (joules) using the following formula:

E = (2.5 x 104) x 101.5M

exponential

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

log10E= 4.4 + 1.5M

logarithmic

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

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

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

KEY: How does sound compare to softest audible sound?

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?

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

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.

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?

Pages 554 - 555

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