Section 8DSection 8DLogarithmic Logarithmic

Scales: Scales: Earthquakes, Earthquakes,

Sounds, & AcidsSounds, & AcidsPages 519-526Pages 519-526

Logarithmic 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.

Successive numbers on the scale increase by the same relative amount.

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

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Earthquakes – Relative Earthquakes – Relative EnergyEnergy

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Relative Energy7.2000E+176.0000E+174.8000E+173.6000E+172.4000E+171.2000E+170

Dotplot of Relative Energy

Each symbol represents up to 3 observations.

Magnitude Scale Magnitude Scale Category Magnitud

eApproximate number

per year(Worldwide average

since 1900)

GreatGreat 8 and up8 and up 11

MajorMajor 7-87-8 1818

StrongStrong 6-76-7 120120

ModeratModeratee

5-65-6 800800

LightLight 4-54-5 60006000

MinorMinor 3-43-4 50,00050,000

Very Very minorminor

Less Less than 3than 3

1,000 / 8,000 per 1,000 / 8,000 per dayday

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Earthquakes – Relative Earthquakes – Relative EnergyEnergy

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Magnitude9.08.47.87.26.66.05.4

Dotplot of Magnitude

The Earthquake Magnitude The Earthquake Magnitude ScaleScale

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The scale is designed so that each magnitude (M) represents about 32 times as much energy as the prior magnitude.

1.525,000 10 ME joules

10log 4.4 1.5E M

no units on magnitude

Examples:Examples:8-D

Sumatra: Dec. 26, 2004magnitude = 9 283,106 deaths

Mexico earthquake: Sept. 19, 1985 magnitude = 8 9,500 deaths

Since each magnitude increase (of 1) means Since each magnitude increase (of 1) means approximatelyapproximately 32 times as much energy- 32 times as much energy-

The December Sumatra released about 32 The December Sumatra released about 32 times as much energy as the 1985 Mexico times as much energy as the 1985 Mexico earthquake, and resulted in almost 30 times earthquake, and resulted in almost 30 times as many deaths.as many deaths.

The Earthquake Magnitude The Earthquake Magnitude ScaleScale

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Where is the ‘almost 32 times as much energy’ coming from?

Ah ha!

1.525,000 10 ME 1.510 31.6227766...

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New Guinea earthquake (June 25, 1976): magnitude = 7.1 energy = 1.1167×1015

joules

# deaths = 422Afghanistan earthquake (May 30, 1998):

magnitude = 6.9 energy = 5.5968×1014 joules

# deaths = 4000 Energy New GuineaEnergy New Guinea = = 1.11671.1167××10101515 = 1.995= 1.995Energy Afghanistan 5.5968Energy Afghanistan 5.5968××10101414

New Guinea earthquake was New Guinea earthquake was about twiceabout twice as as strong as the Afghanistan earthquake.strong as the Afghanistan earthquake.

Another way:Another way:8-D

New Guinea earthquake: 7.1 magnitudeAfghanistan earthquake: 6.9 magnitude

Difference in magnitude = 7.1-6.9 = .2Difference in magnitude = 7.1-6.9 = .2

1.5 .3(.2)10 10 1.99526...

1.525,000 10 ME

Measuring SoundMeasuring Sound The The decibel scaledecibel scale is used to compare is used to compare

the the loudness of sounds.loudness of sounds.

Designed so that Designed so that 0 dB0 dB represents the represents the softest sound audible to the human softest sound audible to the human ear.ear.

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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 10 and intensity is multiplied by 10.

Measuring SoundMeasuring Sound

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

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10

# times louder loudness in dB = 10 log

than softestaudiblesound

or

loudness in dB

10 # times louder 10

thansoftestaudiblesound

ExampleExample

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What is the loudness, in dB, of a sound 25 million times as loud as the softest audible sound?

10

# times louder loudness in dB = 10 log

than softestaudiblesound

ExampleExample

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What is the loudness, in dB, of a sound 25 million times as loud as the softest audible sound?

dB 10 log(25,000,000)

74dB

10

# times louder loudness in dB = 10 log

than softestaudiblesound

ExampleExample

47 13

3.410intensityof sound 110 10

intensity of sound 2

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How much more intense is a 47-dB sound than a 13-dB sound?

loudness of sound 1-loudness of sound 2

10intensityof sound 110

intensity of sound 2

2,512 times more intense

pH ScalepH ScaleThe The pHpH scale is defined by the scale is defined by the

following following equivalent formulas:equivalent formulas:

pH = pH = loglog1010[H[H++] or [H] or [H++] = ] = 1010pHpH

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

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Hydrogen concentration:Hydrogen concentration:8-D

A mole is Avogadro’s number of

particles

= 6×1023 particles

So [H+] is measured in number of 6×1023 particles per liter

pH ScalepH ScaleThe The pHpH scale is defined by the following scale is defined by the following

equivalent formulas:equivalent formulas:

pH = pH = loglog1010[H[H++] or [H] or [H++] = 10] = 10pHpH

Pure waterPure water is is neutral and has a pH of and has a pH of 7. .

[H[H++] = 10] = 107 7 = .0000007 moles/liter= .0000007 moles/literAcids have a pH have a pH lower than than 7Bases (alkaline solutions) have a pH (alkaline solutions) have a pH

higher than than 7..

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Typical pH valuesTypical pH values8-D

SolutionSolution pHpH SolutionSolution pHpH

Pure Pure waterwater

77 Drinking waterDrinking water 6.5-6.5-88

Stomach Stomach acidacid

2-32-3 Baking sodaBaking soda 8.48.4

VinegarVinegar 33 Household Household ammoniaammonia

1010

Lemon Lemon juicejuice

22 Drain openerDrain opener 10-10-1212

ExampleExample 8-D

If the pH of a solution increases from 4 to 6, how much does the hydrogen ion concentration change? Does the change make the solution more acidic or more basic?

Initial concentration = [H1+] = 10-pH

= 10-4 =.0001 moles/literNew concentration = [H2

+] = 10-pH

= 10-6 = .000001 moles/liter

So it decreases by a factor of .0001 = 10-4 = 100

.000001 10-6

ExampleExample 8-D

If the pH of a solution increases from 4 to 6, how much does the hydrogen ion concentration change? Does the change make the solution more acidic or more basic?

Pure water is neutral and has a pH of 7. Acids have a pH lower than 7Bases have a pH higher than 7.

This makes the solution more basic (less acidic).

ExampleExample8-D

How much more acidic is acid rain with a pH of 3 than ordinary rain with a pH of 6?

We really want to know – how many times larger is the hydrogen concentration of the acid rain than that of ordinary rain?

Which means we need to look at the ratio of their hydrogen concentrations:

ExampleExample8-D

How much more acidic is acid rain with a pH of 3 than ordinary rain with a pH of 6?

Ordinary rain: [H+] = 10-pH = 10-6 mole per literAcid rain: [H+] = 10-pH = 10-3 mole per liter

Ratio: 10-3 = 1000 10-6

That is, this acid rain is 1000 times more acidic than ordinary rain.

Homework:Homework:

Pages 526-527Pages 526-527

# 10, 12, 16, 19, 20, 26, 28, 34# 10, 12, 16, 19, 20, 26, 28, 34

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