section 8d logarithmic scales: earthquakes, sounds, & acids pages 519-526
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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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