# You Gotta Know These Scientific Scales

Science requires lots of comparisons. To make comparisons easier, scientists have devised many scales over time. Broadly speaking, the most common scales can be divided into four categories: linear scales, logarithmic scales, power scales, and empirical scales.

## Linear scales

Linear scales are based on straight lines: equal differences between values on the scale indicate equal differences in the phenomenon being described.

• Temperature scales: The scales most frequently used to measure temperatures in science are the Kelvin and Celsius scales. The Celsius scale, developed by Swedish scientist Anders Celsius in the early 1700s, assigns a value of 0°C to the freezing point of water (at a pressure of 1 atmosphere), and 100°C to the boiling point of water (at the same pressure). The Kelvin scale is based on the triple point of water, the point at which water’s solid, liquid, and gaseous phases can coexist in equilibrium: 1 K is defined as 1/273.16 of the temperature of water at its triple point.

Kelvins are treated like all other SI units: a temperature of 100 K is read as “one hundred kelvins,” not “one hundred degrees kelvin.”

Differences on the Celsius scale have the same magnitude as differences on the Kelvin scale: a gap of 1°C between temperatures is the same as a gap of 1 K. Therefore the lowest possible temperature, absolute zero (0 K) is equal to –273.15°C, and (at a pressure of 1 atmosphere) water freezes at 273.15 K and boils at 373.15 K.

• Mach number: Mach numbers measure speed. Based on a suggestion by Swiss aeronautical engineer Jakob Ackeret, it was named after Ernst Mach, an Austrian physicist who studied — among other things — the Doppler effect, sensory perception, and the origin of inertia. The Mach number is defined as the ratio of the speed of an object to the speed of sound in the same medium. So an object moving at the speed of sound (which in dry air at 20°C is about 343 m/s) has a speed of Mach 1, and an object moving at twice the speed of sound has a speed of Mach 2.

## Logarithmic scales

Logarithmic scales are based on the concept of logarithms: a gap of one unit between measurements on the scale always corresponds to the same ratio between the phenomena being described.

• Decibel scale: The decibel scale can describe any kind of power, but is most commonly used to describe the intensity of sound waves. The decibel (abbreviated dB) is one tenth of a larger unit, the bel (abbreviated B), named after inventor Alexander Graham Bell. The intensity of a sound in decibels is given by the formula I = 10 log(P1/P0), where P1 is the intensity of the sound being measured (in watts per square meter) and P0 is a reference intensity, which is based on the least powerful sound wave that can be detected by the average human ear (namely 10–12 W/m2). A 10-dB increase in sound intensity corresponds to multiplying the energy of the sound wave by 10. A normal conversation has a volume of about 70 dB.
• pH scale: The pH scale, developed by S. P. L. Sørensen in 1909, is used to quantify acidity. The pH (power of hydrogen) of a solution is defined as the opposite of the (base-10) logarithm of the concentration of protons in a solution: pH = –log10[H+]. (The brackets are standard notation in chemistry for “the concentration of.”) Thus, greater concentrations of protons correspond to smaller pH values. At 25°C, the neutral (neither acidic nor basic) pH is 7; solutions with a pH less than 7 are considered acidic, and solutions with a pH greater than 7 are considered basic.
• Richter scale: The Richter scale measures earthquake intensity. Developed by Caltech professor Charles Francis Richter, it measures the shaking intensity associated with earthquakes, as quantified by the amplitude of vibrations on a seismograph. A magnitude-5.0 earthquake will have an amplitude 10 times larger than that of a magnitude 4.0 quake. The energy associated with an earthquake is actually proportional to the 3/2 power of the magnitude: a 1-point difference on the Richter scale corresponds to a 103/2-fold (about 31.6) difference in energy. Because of difficulties in measuring the magnitudes of large earthquakes, the Richter scale has been superseded by the moment magnitude scale, which uses a different formula but retains the logarithmic nature (and correlates to measurements on the Richter scale).

## Power scales

While logarithmic scales take the logarithm of the measured number, power scales raise it to an exponent. These two scales happen to use an exponent of 3/2:

• Beaufort wind force scale: The first official use of the Beaufort scale was on the voyage of the HMS Beagle in 1831, which was led by Robert FitzRoy, who had been trained by the scale’s namesake, Sir Francis Beaufort, a rear admiral in the British Navy. The Beaufort scale is primarily based on wind speed, but also incorporates descriptions of wave height, sea conditions, and land conditions. It starts at 0, corresponding to “calm” winds with a speed less than 1 knot. In most parts of the world, it stops at 12, which is designated “hurricane-force winds.” Scores of 2 to 6 are called “breezes,” and scores of 7 to 10 are called “gales.” Since 1946 the median wind speed for each step has been defined as 1.87 B3/2, where B is the number on the Beaufort scale.
• Fujita-Pearson and Enhanced Fujita scales: These scales measure tornado strength. The Fujita-Pearson scale was introduced in 1971 by Ted Fujita, a meteorology professor at the University of Chicago, and Allen Pearson, director of what is now the Storm Prediction Center. The original scale went from F0 (wind speeds lower than hurricane force, which should cause little to no damage) to F5 (wind speeds of 260 miles per hour, which should cause “incredible damage”). Fujita also included an “inconceivable damage” category for tornados exceeding 319 miles per hour (theoretically possible, but no such tornado has ever been observed). In 2007 the Enhanced Fujita (EF) scale was introduced, narrowing the speed ranges for each category: an EF5 begins at “just” 200 miles per hour.

## Empirical scales

These scales assign numbers that make sense relative to each other, but are arbitrary in the sense that there is no fixed quantitative relationship between values on the scale.

• Mohs scale: Invented by Friedrich Mohs in 1812, the Mohs scale is based on the abilities of minerals to scratch one another. The original scale assigned a value of 1 to talc, which can be scratched by essentially every solid known, and a value of 10 to diamond, which (among naturally occurring minerals) can only be scratched by other diamonds. The steps are of arbitrary size, with a 1-point difference corresponding to anywhere from a 1.5-fold to a 4-fold increase in hardness; diamond is about 1600 times harder than talc. Following the discovery of more ultra-hard minerals, scientists have proposed extending the scale so that diamond is 15 instead of 10.)
• Pauling scale: The Pauling scale, devised by Linus Pauling in 1932, is one of several scales that measure electronegativity, the extent to which atoms attract electrons in chemical bonds. Higher values correspond to stronger attractions. Francium has the lowest value, about 0.7, while fluorine has the highest value, about 4.0. (Noble gases are not assigned values on the Pauling scale, since they were not known to form any bonds when Pauling devised it.) Differences in electronegativity characterize bonds: the greater the difference, the more ionic the bond.
• Saffir-Simpson scale: The Saffir-Simpson scale is a measure of wind speed and damage from hurricanes. It was developed in 1971 by Herbert Saffir, a civil engineer, and Robert Simpson, then the director of the National Hurricane Center. It rates hurricanes on a 1-to-5 scale: a 1 corresponds to a wind speed of 74 to 95 miles per hour, which causes “some damage.” A 5 causes “catastrophic damage,” with wind speeds over 157 miles per hour and affected areas uninhabitable for weeks or months.