Finally, is a trillionth of a milligram really 10−21 mg? A trillion has twelve zeros, not twenty-one. Actually it is the same because Loschmidt was using the long scale for his numbers. He was writing in Austria where a billion is a million million, and a trillion is a million billion, with eighteen digits.
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Finally we can turn to one of the really big numbers in science, Avogadro’s number. Avogadro made the statement that equal volumes of different types of gas contained the same number of molecules. But we do not define his number in terms of volume. Avogadro’s number is a number we use for counting out stuff. It is like using the word “dozen” to mean you counted out twelve, or a “gross” to mean one hundred and forty-four (a dozen dozen). If you count out an Avogadro number of atoms or molecules you have a mole of that stuff. Now one mole of any gas under normal conditions does occupy the same volume. One mole of gas occupies 22.4 l, or about 6 gallon jugs. But gas can be heated such that it expands, or chilled to a liquid, or even frozen to a solid, and still a mole will contain the same number of atoms or molecules. So whereas we recognize Avogadro was right at a set temperature and pressure (one mole is 22.4 l at 0°C and at a normal atmospheric pressure), we now define a mole and Avogadro’s number in terms of mass. Twelve grams of carbon-12 is exactly (by current definitions) one mole of that substance, and contains exactly Avogadro’s number of carbon atoms.
What about a mole of water? This will be important in answering our wine glass question, since most of wine is made up of water. Water is chemically H2O, two moles of atomic hydrogen weigh about 2 g, whereas one mole of atomic oxygen weighs about 16 g. So a mole of water is about 18 g, or we can say that our wine glass holds about 5.6 moles of water.
But how many atoms in a mole? Loschmidt told us the mass of an atom, so we can use his number to find that there are about 1024 atoms in our 12-g chunk of carbon. That is essentially Avogadro’s number.
Actually over the last century we have improved upon Loschmidt’s results a lot. What we do is we take a 12-g chunk of carbon and count the atoms in it. Well, maybe not directly. For example, the Hope diamond is about 9 g, and so three-quarters of a mole. The number of atoms in it is a 24-digit number. What we can do, with a high degree of precision, is measure the space between atoms in a crystal. The measurement of structure and spacing is called crystallography, and we will discuss it in more detail in a later chapter on atomic scales. For now, however, what is important to us is that we can shine an X-ray on a crystal and something like a Moiré pattern of light and dark spots is formed. From measuring these patterns we can determine the spacing of atoms in the crystal very well. But knowing the spacing of carbon atoms in a diamond crystal is not quite the same as knowing how many atoms there are. Counting the number of atoms in a crystal from spacing is a lot like calculating the number of people in a crowded room. If there is about a meter between people at a party, and the room has an area of 40 m2, then there are about 40 people there. We would actually do a much better job of our crowd estimation if people would cooperate and stand in a regular formation. Perhaps instead of studying the party, we could try counting the people in a marching band who are all lined up in rank and file formation. If they are spaced a meter apart, an arm’s length to your neighbor’s shoulder is very close to a meter, and if we know the area the band covers, we know the number of people very well. This regularity is why we turn to crystals of carbon—diamonds—instead of chunks of coal. Diamonds have a regularity that repeats, row after row, column after column, layer after layer, throughout the whole macroscopic crystal.
So now, at long last we come to the famous Avogadro’s number:
NA = 6.02214129 × 1023
molecules or atoms in a mole.
Avogadro never saw this number, but based upon Loschmidt’s reaction to his own results I think Avogadro would have been astonished at its size. Twelve grams of carbon contain 6 × 1023 atoms. The number was named after Avogadro more than a century after he stated his “same number of gas molecules in the same volume” hypothesis. In fact in some places, especially German-speaking countries, the number is referred to as Loschmidt’s number, but I think this is becoming less common.
Now let us turn to our wine glass and ocean problem one last time. There are 5.6 moles of water in 1 dl, or 1 wine glass. That means that there are about 3.3 × 1024 molecules in our wine glass. From our previous calculation we find that there are about 1.3 × 1022 glasses of water in all of the oceans. To answer Lord Kelvin’s original question, “If I pour a glass of wine in the ocean and stir the ocean completely, how many of the original wine molecules will be in a cup of water I scoop out of any sea?” The answer is the ratio of the number of molecules in a cup to the number of cups in the ocean,
There will be about 250 molecules of the original wine in every cup scooped out of any ocean, or any other water anywhere on Earth. This result is amazing. It not only means we can we measure atoms and calculate their number, but it also says that each and every cup is full of history. Given time, the oceans do stir themselves up, and water molecules tend to be stable across thousands and millions of years. That means that my cup of water, dipped out of the Connecticut River contains some of Lord Kelvin’s wine. It also contains some of the wine used to launch Cleopatra’s barge, as well as any other historic event you can think of. It contains some of the Lake Erie water Governor Dewitt Clinton poured into New York harbor to mark the opening of the Erie Canal in 1825. It also contains some of the wine Homer sipped as he composed the phrase “wine dark sea.”
There are about 8 × 1045 molecules of water on Earth. Later we will look at how many atoms and even quarks there are in the universe, and maybe we will start to understand what “a drop in the ocean” really means.
4
Scales of Nature
Imagine sitting in a wind that is roaring along at scores of kilometers per hour, yet not having a hair blown out of place. This may seem absurd, but that is the experience of high altitude balloonist. When one crosses the Atlantic in a gondola suspended beneath a balloon, the winds at high altitude may rush you along but you can set papers on a table and not worry about them being blown away. This is because you, the air, the gondola, the balloon and the papers are all moving together.
Measuring wind speed turns out to be a bit more tricky than it might first appear, because we cannot directly see the wind. But once we realize that balloons move with the wind, we have a handle to tackle this problem. So now before we go and measure the universe I would like to pause and talk about how we measure some simple things, like wind speed, starlight and the hardness of rocks. Actually it is not the measurement techniques that interest us here, but rather the scale upon which we plot these measurements. These scales are called logarithmic, but before we get side tracked on what that means, let us return to those balloons.
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A balloon moves with the wind once it has been untethered from the Earth, and tracking a balloon is relatively easy. So by measuring the velocity of the balloon we can measure the velocity of the wind. Even today, NASA will release balloons before a rocket launch and the National Weather Service will launch dozens of balloons daily. Not only can you track a balloon and measure wind speed, but the balloon tends to keep rising and so you can measure the speed at various altitudes. But quite honestly, it is not the simplest technique, and it tends to use up a lot of balloons. An alternative method is to use a device called an anemometer.
You have probably seen an anemometer. It is a simple wheel with cups designed to catch the wind and spin around. Typically the wind catchers are three or four hemispheres mounted on spokes around a vertical axis. The open side of the cups catch the wind and are pushed back. The other side of the cups are smooth and offer little resistance to the air, and so can be pushed into the wind, letting the wheel spin around. By measuring the rotation rate of the wheel one can measure the wind speed. This simple device was invented by Dr. John Thomas Romney Robinson of the Armagh Observatory in N
orthern Ireland in the 1840s. It is now such a common thing that it can be seen in schools and at airports. It is used at both professional and amateur weather stations everywhere. It is ubiquitous, but it is not the only way to measure wind speed, and the need to measure wind preceded the anemometer.
Historically people would describe the wind in terms like a “stiff breeze,” or a “moderate wind,” or a “gale.” But the description of wind was inconsistent until Sir Francis Beaufort of the British Royal Navy established what we now call the Beaufort scale. In 1805–6, Beaufort was commanding HMS Woolwich, conducting a hydrographic survey off of the Río de la Plata in South America. He first developed his scale as both a shorthand notation for the wind conditions, as well as a set of standards to make his reports consistent. For example, if he had just enough wind to steer by, he called it a “light air,” and marked it as a 1 on his scale. If he had enough wind so he could sail at five or six knots he called it a “moderate breeze,” and recorded it as a 4.
His system might have remained a private shorthand, but a sniper wound cut his sea career short and led him to a land-locked position at the British Admiralty. Here, he was eventually appointed Hydrographer to the Admiralty, and by 1838 the Beaufort wind force scale was the standard way that all ships in the Royal Navy recorded wind conditions.
The Beaufort scale has evolved over time, but it has always been based on the idea that the number reported is derived from an effect that can be seen. Originally all the standards were things seen on sailing ships. If a ship needed to double reef its sails, the wind was a Beaufort 7. But ships evolved away from sails and sailing conditions, and so the scale was adopted to the conditions of the sea. A Beaufort 7 is called a “moderate gale,” and on the ocean we see that “the seas heap up and foam begins to streak.” On land you can recognize a 7 because “whole trees are in motion and an effort is needed to walk against the wind.” What is missing from our twenty-first century point of view is an objective, absolute scale that would somehow allow us to relate Beaufort’s scale to kilometers per hour. But before that let us look at extreme winds.
An extension of the Beaufort scale is the Saffir–Simpson hurricane scale. This is the scale that is used when we hear on the news that “Hurricane Katrina has been upgraded to a category 5,” or “Sandy is a category 1.” This scale is similar to the Beaufort scale in that it is based upon the expected effects of the storm. In a category-2 hurricane, one can expect the loss of roofs and damage to mobile homes and poorly constructed structures. In a category 5, such as Katrina in 2005, most trees are blown over and most manmade structures can expect major damage.
In about 1970, Herbert Saffir, a structural engineer, was studying hurricane damage for the UN. He developed the scale to help prepare for the aftermath of storms. If a category 3 is coming ashore, relief agencies will already know approximately how much damage to expect and how many resources they should be ready to deploy. Later Bob Simpson added the effects of the storm surge to the damage calculation, which makes a lot of sense, since hurricanes that are formed over the seas do most of their damage when they make their landfall. Actually, since 2010 the scale has been redefined to not include the storm surges, the reason cited being that these effects are hard to calculate and are often very local.
A curious trait of both the Beaufort and Saffir–Simpson scales is that these scales are primarily about the effects of the wind and not the wind speed. But the two measurements can be related. I can hold up my hand held anemometer and measure the wind speed under various Beaufort conditions. The translation has now been standardized as:
Here v is the velocity of the wind and B is the number on the Beaufort scale. This equation is worth dissecting because so much of nature has a similar form. The “0.836 [m/s]” means that we are translating from the Beaufort scale to a velocity measure in meters per second. If we wanted miles per hour or kilometers per hour there would have been a different constant here. It is the 3/2 that is really interesting. It tells us that if the Beaufort number goes up by four, the velocity has gone up by eight. The velocity changes faster than the Beaufort number. If I double the wind speed, the effect is less than doubled. As the equation is written above we would say that the “velocity rises exponentially with the Beaufort number.” We could turn around the equation, and write the effect as a function of the wind speed as:
B = log3/2(v)
which tells us that the physical effect (B) rises logarithmically with velocity (v) (see Figure 4.1).
Figure 4.1 The Beaufort and Saffir–Simpson scales and windspeed. The Beaufort and Saffir–Simpson scales are a quantification of the effects of the wind. We see that the resulting effects rise slower then the actual wind speed.
The fact that the effects that are felt change more slowly than the quantities that we measure pervades all aspects of nature. In Chapters 1 and 2 there was a graph that related the size of a wide range of objects such as atoms, cells, planets and stars by plotting them on a logarithmic scale. But when you looked at those plots you do not see the word logarithm anywhere. What you do see is that for every step to the right the sizes get bigger by a factor of ten and every step to the left they get smaller by that same factor. We could have written 1, 10, 100, 1000, 10000 and so forth, but instead we wrote 100, 101, 102, 103, 104, using scientific notation. Now every increase is just a step of one in the exponent. For us it was a convenient way to relate the size of neutrons, whales and galaxies on the same graph. But there is more to it than just convenience. Nature really is this way. Phenomena change more slowly than sizes, distances, time, or energy.
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Another scale that measures the destructive effect of a natural phenomenon is the Richter scale of earthquakes. Us humans are always wanting to compare things and so a very natural question is “was this earthquake I just felt bigger than the one someone else felt a year ago?” Before I build a “bigness”-measuring device I had really best decide what that means. I could argue that since an earthquake is all about motion, energy has been released, and therefore I would like to compare the energies of different earthquakes. But measuring that energy is not so simple and in the end it might not be the best way to describe an earthquake as it relates to humans.
Much like the Beaufort scale, the Richter scale arose out of a technique of measurement. Charles Richter and Beno Gutenberg were researchers at the Carnegie Institute and Caltech where, in 1935, they were using a Wood–Anderson seismograph to measure earthquakes. This was a standard device at the time. In it a paper chart slowly rolled past a suspended pen. When the ground shook, the pen would quiver and sway and leave a squiggle on the paper. Now the problem facing people trying to interpret seismic data was that different reporting stations would measure different size squiggles for the same quake. The reason was obvious: some stations were nearer to the epicenter, the source of the earthquake, than others. Richter developed the local magnitude scale—commonly called the Richter scale—as a method of adjusting for this. Richter combined the distance from the epicenter and the measurement of the seismograph to determine the magnitude of the earthquake with his equation:
ML = log(A) + 3 log(d) − 2.92
Here A is the amplitude of the squiggle measured on the chart and d is the distance from the epicenter. What this means is that if you are a standard distance from the epicenter (100 km) a magnitude 1, 2, or 3 earthquake will set your pen swaying by 1, 10, or 100 micrometers. A magnitude 5 will set the pen swaying by 105 micrometers, which is 10 cm or 4 inches.
The Richter scale has been supplanted by other seismic scales for a working geologist, but it still thrives in the public consciousness. It is still a scale we talk about and it is the way that the magnitude of a quake is reported in the news. So how big is a magnitude 1 or 4 or 8? A magnitude 1 earthquake can be caused by 32 kg of TNT, a typical blast at a construction site. A magnitude 2 could be caused by a ton of TNT, which is a large blast at a quarry. Going from 32 kg to a 1000 kg in one step of this seismic scale, we can s
ee that the energy—the amount of TNT—which is needed to cause an earthquake one step greater is rising much faster than the Richter magnitude. The great earthquake of San Francisco of 1906 was a magnitude 8.0. The largest earthquake ever recorded, the Chilean earthquake of 1960, was a 9.5.
As a side note, another interesting seismic scale is the Mercalli intensity scale, named after Giuseppe Mercalli (1850–1914). Mercalli was a priest and professor of the natural sciences who studied volcanology in Italy. His scale was based on the effects felt by people. For example, a Mercalli intensity II would be felt by people, especially people in the upper floors of a building. It is equivalent to a Richter 2 or 3. An intensity VII would cause some damage in well-built buildings and major damage to poorly built ones, and is equivalent to a Richter 6. What is so interesting to me about the Mercalli scale is that it is based purely upon the experience of people near the quake. Moreover, the Mercalli scale tracks almost exactly with the Richter scale (see Figure 4.2), which is based upon the squiggles of a shaking pen or, more precisely, upon the logarithm of the size of the squiggles of that pen or the logarithm of the energy of the quake.
Finally, we can use the magnitude of the quake, either on the Richter scale or the Mercalli scale, to estimate the resulting damage. This is a lot like the way we use the Saffir–Simpson hurricane scale. The US Geological Survey tells us that magnitude 3 earthquakes are “often felt, but rarely cause damage,” whereas a magnitude 8 “can cause serious damage in areas several hundred miles across.”
Figure 4.2 The Richter scale versus the Mercalli scale. The Richter scale (based on the logarithm of the pen motion, or energy) tracks with the Mercalli scale, which is based on human experience.
How Big is Big and How Small is Small Page 6