Notes 8D Logarithmic Scales: Earthquakes, Sounds, and Acids
Table 8.4 Earthquake Categories and Their Frequency
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
magnitude 2–3: 1000 per day
magnitude 1–2: 8000 per day
THE EARTHQUAKE MAGNITUDE SCALE
The magnitude scale for earthquakes is defined so that each magnitude represents about 32 times
as much energy as the prior magnitude. More technically, the magnitude, M, is related to the
released energy, E, by the following equivalent formulas:
log 10 E  4.4  1.5M
or
E  (2.5  10 4 )  101.5 M
The energy is measured in joules (see Unit 2B); magnitudes have no units.
Example 1 The Meaning of One Magnitude
Using the formula for earthquake magnitudes, calculate precisely how much more energy is
released for each 1 magnitude on the earthquake scale.
Example 2 Comparing Disasters
The 1989 San Francisco earthquake, in which 90 people were killed, had magnitude 7.1.
Calculate the energy released, in joules. 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.
Table 8.5 Typical Sounds in Decibels
Times as Loud
as Softest
Decibels Audible Sound
Example
14
140
10
jet at 30 meters
120
1012
strong risk of damage to human ear
10
100
10
siren at 30 meters
9
90
10
threshold of pain for human ear
80
108
busy street traffic
6
60
10
ordinary conversation
40
104
background noise in average home
2
20
10
whisper
10
101
rustle of leaves
0
1
softest audible sound
-10
0.1
inaudible sound
THE DECIBEL SCALE FOR SOUND
The loudness of a sound in decibels is defined by the following equivalent formulas:


intensity of the sound

loudness in dB  10log 10 
 intensity of softest audible sound 
or
intensity of the sound
 10 loudnessin dB/10
intensity of softest audible sound
Example 3 Computing Decibels
Suppose a sound is 100 times as intense as the softest audible sound. What is its loudness, in
decibels?
Example 4 Sound Comparison
How does the intensity of a 57-dB sound compare to that of a 23-dB sound?
THE INVERSE SQUARE LAW FOR SOUND
The intensity of sound decreases with the square of the distance from the source, meaning that
the intensity is proportional to 1/d2. We therefore say that sound follows an inverse square law
with distance.
Example 5 Sound Advice
How many times greater is the intensity of sound from a jet at a distance of 30 meters than the
intensity at a distance of 40 meters?
The pH Scale for Acidity
If you check the labels of many household products, including cleansers, drain openers, and
shampoo, you will see that they sate a quantity called the pH. The pH is used by chemists to
classify substances as neutral, acidic, or basic (also called alkaline). By definition,
 Pure water is neutral and has a pH of 7
 Acids have a pH lower than 7
 Bases have a pH higher than 7
Table 8.6 give a few typical pH values.
Table 8.6
Solution
lemon juice
stomach acid
vinegar
drinking water
pH
2
2–3
3
6.5
Solution
pH
pure water
7
baking soda
8.4
household ammonia 10
drain opener
10–12
Chemically, acidity is related to the concentration of positively charged hydrogen ions, which are
hydrogen atoms with out their electron. The ions themselves are denoted H+ for hydrogen with a
positive charge. The concentration of hydrogen ions is denoted [H+] and is usually measured in
units of moles per liter. A mole is Avogadro’s number of particles, which is about 6 x 1023.
THE PH SCALE
The pH scale is defined by the following equivalent formulas:
pH  -log10 [H  ]
Where [H+] is the hydrogen ion concentration in moles per liter.
Example 6 Finding pH
What is the pH of a solution with a hydrogen ion concentration of 10-12 mole per liter?
Example 7 Acid Rain versus Normal Rain
In terms of hydrogen ion concentration, compare acid rain with a pH of 2 to ordinary rain with a
pH of 6.
Example 8 Find the Hydrogen Ion
What’s the hydrogen ion concentration of a substance with a pH of 8.9?