Transcript Section 3B Putting Numbers in Perspective

Section 8D
Logarithmic Scales:
Earthquakes, Sounds, &
Acids
Pages 519-526
8-D
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.
8-D
Earthquakes – Relative Energy
Dotplot of Relative Energy
0
1.2000E+17 2.4000E+17 3.6000E+17 4.8000E+17 6.0000E+17 7.2000E+17
Relative Energy
Each symbol represents up to 3 observations.
8-D
Magnitude Scale
Category
Magnitude
Approximate number per year
(Worldwide average since 1900)
Great
8 and up
1
Major
7-8
18
Strong
6-7
120
Moderate
5-6
800
Light
4-5
6000
Minor
3-4
50,000
Very minor Less than 3
1,000 / 8,000 per day
8-D
Earthquakes – Relative Energy
Dotplot of Magnitude
5.4
6.0
6.6
7.2
Magnitude
7.8
8.4
9.0
8-D
The Earthquake Magnitude Scale
The scale is designed so that each
magnitude (M) represents about 32 times
as much energy as the prior magnitude.
E  25,000 10
1.5 M
joules
log10  E   4.4  1.5M
no units on magnitude
8-D
Examples:
Sumatra: Dec. 26, 2004
magnitude = 9
283,106 deaths
Mexico earthquake: Sept. 19, 1985
magnitude = 8
9,500 deaths
Since each magnitude increase (of 1) means
approximately 32 times as much energy-
The December Sumatra released about 32
times as much energy as the 1985 Mexico
earthquake, and resulted in almost 30 times
as many deaths.
8-D
The Earthquake Magnitude Scale
Where is the ‘almost 32 times as much
energy’ coming from?
E  25,000 10
1.5 M
10  31.6227766...
1.5
Ah ha!
8-D
New Guinea earthquake (June 25, 1976):
magnitude = 7.1 energy = 1.1167×1015 joules
# deaths = 422
Afghanistan earthquake (May 30, 1998):
magnitude = 6.9
energy = 5.5968×1014 joules
# deaths = 4000
Energy New Guinea = 1.1167×1015 = 1.995
Energy Afghanistan
5.5968×1014
New Guinea earthquake was about twice as
strong as the Afghanistan earthquake.
8-D
Another way:
New Guinea earthquake: 7.1 magnitude
Afghanistan earthquake: 6.9 magnitude
Difference in magnitude = 7.1-6.9 = .2
1.5(.2)
10
 10  1.99526...
.3
E  25,000 10
1.5 M
8-D
Measuring Sound

The decibel scale is used to compare the
loudness of sounds.

Designed so that 0 dB represents the softest
sound audible to the human ear.
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.
8-D
Measuring Sound

The loudness of a sound in decibels is defined
by the following equivalent formulas:

# times louder

 10
thansoftest audiblesound
loudness in dB 

10

or
# times louder


loudness in dB = 10  log10 

than
softest
audible
sound


8-D
Example
What is the loudness, in dB, of a sound 25
million times as loud as the softest
audible sound?
# times louder


loudness in dB = 10  log10 

than
softest
audiblesound


8-D
Example
What is the loudness, in dB, of a sound 25
million times as loud as the softest
audible sound?
# times louder


loudness in dB = 10  log10 

than
softest
audiblesound


dB  10  log(25,000,000)
 74dB
8-D
Example
How much more intense is a 47-dB sound
than a 13-dB sound?
 loudness of sound 1-loudness of sound 2 


10


 intensityof sound 1 

  10
 intensity of sound 2 
4713
 intensityof sound 1 
3.4
10

10

10


 intensity of sound 2 
 2,512 times more intense
8-D
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.
Hydrogen concentration:
A mole
is Avogadro’s number of particles
= 6×1023 particles
So [H+] is measured in number of
6×1023 particles per liter
8-D
8-D
pH Scale
The pH scale is defined by the following
equivalent formulas:
pH = log10[H+] or [H+] = 10pH
Pure water is neutral and has a pH of 7.
[H+] = 107 = .0000007 moles/liter
Acids have a pH lower than 7
Bases (alkaline solutions) have a pH higher
than 7.
8-D
Typical pH values
Solution
Pure water
Stomach acid
pH
7
2-3
Solution
pH
Drinking water
6.5-8
Baking soda
8.4
10
Vinegar
3
Household ammonia
Lemon juice
2
Drain opener
10-12
Example
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/liter
New concentration = [H2+] = 10-pH
= 10-6 = .000001 moles/liter
So it decreases
by a factor of .0001 = 10-4 = 100
.000001 10-6
Example
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 7
Bases have a pH higher than 7.
This makes the solution more basic (less acidic).
Example
8-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:
Example
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 liter
Acid 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.
8-D
8-D
Homework:
Pages 526-527
# 10, 12, 16, 19, 20, 26, 28, 34