Section 8DLogarithm Scales: Earthquakes, Sounds, and Acids

Pages 519-528

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.

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

Earthquake Magnitude Scale

The magnitude M is related to the released energy E by the following formulas:

log10(E) = 4.4 + 1.5M or E = (2.5 x 104) x 101.5M

Energy is measured in joules.Magnitudes have no units.

ex1/521 Calculate 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.

ex2/521 How much energy was released by the magnitude 7.1 earthquake in San Francisco in 1989? by the magnitude 6.9 earthquake in Afghanistan in 1998?

SF in 1989 with M=7.1: E = (2.5 x 104) x 101.5(7.1)

= 1.11671E15 joules

Afghanistan in 1998 with M=6.9: E = (2.5 x 104) x 101.5(6.9) = 5.5968E14 joules

Since 1.11671E15/5.5968E14 = 1.99526, we say that:

the SF quake was 1.99526 times more powerful than the Afghanistan quake.

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.

Decibel Scale for Sound

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

10loudness in dB = 10 log ( )intensity of thesoundintensityof softest audible sound

1010loudness in dBintensity of sound

intensity of softest audible sound

More Practice

15/527 How many times as loud as the softest audible sound is the sound of busy street traffic?

17/527 What is the loudness, in decibels, of a sound 45 million times as loud as the softest audible sound?

19/527 How much louder (more intense) is a 35-dB sound than a 10-dB sound?

pH Scale for Acidity

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

Solution pH Solution pH

Pure water 7 Drinking water 6.5-8.2

Stomach acid 2-3 Baking soda 8.4

Vinegar 3 Household ammonia

10

Lemon Juice 2 Drain opener 10-12

The 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.

Practice

27/527 What is the hydrogen ion concentration of a solution with pH 8.5?

29/527 What is the pH of a solution with a hydrogen ion concentration of 0.1 mole per liter? Is this solution an acid or base?

Homework

Pages 526 - 527# 12,14, 18,20, 28,30