Logarithmic Scales 37

Transcription

1 37

2 When a physical quantity varies over a very large range, it is often convenient to take its logarithm in order to have a more manageable set of numbers. We discuss three such situations: the ph scale, which measures acidity; the Richter scale, which measures the intensity of earthquakes; and the decibel scale, which measures the loudness of sounds. Other quantities that are measured on logarithmic scales are light intensity, information capacity, and radiation. 38

3 The ph Scale Chemists measured the acidity of a solution by giving its hydrogen ion concentration until Søren Peter Lauritz Sørensen, in 1909, proposed a more convenient measure. He defined where [H + ] is the concentration of hydrogen ions measured in moles per liter (M). He did this to avoid very small numbers and negative exponents. 39

4 For instance, if [H + ] = 10 4 M, then ph = log 10 (10 4 ) = ( 4) = 4 Solutions with a ph of 7 are defined as neutral, those with ph < 7 are acidic, and those with ph > 7 are basic. Notice that when the ph increases by one unit, [H + ] decreases by a factor of

5 Example 8 ph Scale and Hydrogen Ion Concentration (a) The hydrogen ion concentration of a sample of human blood was measured to be [H + ] = M. Find the ph and classify the blood as acidic or basic. (b) The most acidic rainfall ever measured occurred in Scotland in 1974; its ph was 2.4. Find the hydrogen ion concentration. Solution: (a) A calculator gives ph = log[h + ] = log( ) 7.5 Since this is greater than 7, the blood is basic. 41

6 Example 8 Solution cont d (b) To find the hydrogen ion concentration, we need to solve for [H + ] in the logarithmic equation log[h + ] = ph So we write it in exponential form. [H + ] = 10 ph In this case ph = 2.4, so [H + ] = M 42

7 The Richter Scale In 1935 the American geologist Charles Richter ( ) defined the magnitude M of an earthquake to be where I is the intensity of the earthquake (measured by the amplitude of a seismograph reading taken 100 km from the epicenter of the earthquake) and S is the intensity of a standard earthquake (whose amplitude is 1 micron = 10 4 cm). 43

8 The magnitude of a standard earthquake is M = log = log 1 = 0 Richter studied many earthquakes that occurred between 1900 and The largest had magnitude 8.9 on the Richter scale, and the smallest had magnitude 0. This corresponds to a ratio of intensities of 800,000,000, so the Richter scale provides more manageable numbers to work with. For instance, an earthquake of magnitude 6 is ten times stronger than an earthquake of magnitude 5. 44

9 Example 9 Magnitude of Earthquakes The 1906 earthquake in San Francisco had an estimated magnitude of 8.3 on the Richter scale. In the same year a powerful earthquake occurred on the Colombia-Ecuador border that was four times as intense. What was the magnitude of the Colombia-Ecuador earthquake on the Richter scale? Solution: If I is the intensity of the San Francisco earthquake, then from the definition of magnitude we have M = log =

10 Example 9 Solution cont d The intensity of the Colombia-Ecuador earthquake was 4I, so its magnitude was M = log = log 4 + log = log

11 The Decibel Scale The ear is sensitive to an extremely wide range of sound intensities. We take as a reference intensity I 0 = W/m 2 (watts per square meter) at a frequency of 1000 hertz, which measures a sound that is just barely audible (the threshold of hearing). The psychological sensation of loudness varies with the logarithm of the intensity (the Weber-Fechner Law), so the intensity level B, measured in decibels (db), is defined as 47

12 The intensity level of the barely audible reference sound is B = 10 log = 10 log 1 = 0 db 48

13 Example 11 Sound Intensity of a Jet Takeoff Find the decibel intensity level of a jet engine during takeoff if the intensity was measured at 100 W/m 2. Solution: From the definition of intensity level we see that B = 10 log = 10 log = 10 log = 140 db Thus, the intensity level is 140 db. 49