$44,000,000,000,000.00
or in a slightly different form,
$44 000 000 000 000,00
which reminds us that we need to be careful with our commas, spaces and periods as we move into a global marketplace.
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We can use big numbers to count things like the number of nails in a shipment, or pennies in the world’s economy, or the stars in the sky. This may be useful, but it is not the most interesting aspect of a big number. What is interesting is when we can take a bit of information in the form of a number and combine it with other information to see something new and enlightening. For example, I remember reading a few years ago about a soccer game that was leading up to the World Cup. The Chinese national team was playing and work in China essentially came to a stop during the game. My question is how much work was it that was not done? There are about 1.3 billion (109) people in China. If half of them are working and they stop for a two-hour soccer match, that is the equivalent to 650,000 man-years of labor that was not done. I just combined the number of people, the hours for the game and the hours worked per year.
In a different place I read that some archaeologists and engineers have estimated that the Egyptian pyramids took 30–40,000 people about 20 years to build. That is, it took 600,000 to 800,000 man-years of labor to build the pyramids. That means that one soccer game caused the equivalent of one pyramid to not be built. Although soccer matches and pyramids are both interesting, what I mean to show here is that we will often want to combine these large numbers to create a new insight. I can understand an invisible thing, such as the labor force of a nation or the effect of a large disruption, in terms of the creation of a very visible object like the pyramids.
There is a classic problem that Erwin Schrödinger attributed to Lord Kelvin, which involves some pretty big numbers and which leads to some profound insights about our world. Take a cup of wine and pour it into the ocean. Now wait until the ocean waters have had time to be stirred together, time until some of the wine has seeped into both the Atlantic and the Pacific. Wait until it has diffused into the Mediterranean, under the ice caps of the Arctic and into the warm waters of the Bay of Bengal. Now dip out a cup of water from any ocean or sea on Earth. How many molecules from that original wine glass are in my sample cup?
I have seen this problem used to advance many arguments and points of view such as “Can we tell water molecules apart?,” “Do the deep waters of the ocean ever really mix?” and so forth. But for us this problem is not about quantum identity or mixing; we really just want to look at the number of molecules of wine or water in a cup and the number of cups in the ocean.
We are actually going to work out Lord Kelvin’s problem. Usually I will not write out the details of a calculations, but I will this time, so as to show how numbers flow through a problem. We will solve Lord Kelvin’s problem by first estimating the number of cups in the ocean, using the size of the Earth, the fraction of the Earth covered by water, and the average depth of the oceans.
The Earth is about 40,000 km around, which means it has a surface area of a bit more than 510 million km2. About three-quarters of the surface is covered by water, which is about 380 million km2. Finally the average depth of the ocean is about 3.5 km, so the total volume of water is 1.3 billion km3.
Now each kilometer is a thousand meters on a side, and each cubic meter contains 1000 l, and each liter can fill ten wine glasses. So here is our big number,
That is thirteen sextillion (short scale) or thirteen thousand trillion (long scale). There must be a better way of writing numbers like this because we have not even started writing out the number of molecules.
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So how do we write out really big numbers? We can write out all the digits and take half a line or we can use words like sextillion, thousand-trillion or the more obscure trilliard, but that would send us to our dictionaries to figure out the number of zeros. Instead, we commonly use an alternative: scientific notation. In scientific notation we would write the number of wine glasses of water on Earth as 1.3 × 1022. This convention builds on the idea that 100 = 10 × 10 = 102. The exponent—in the previous sentence the 2—tells us how many tens were multiplied together and how many zeros are in the full number. So I can write a million as 1, 000, 000 = 10 × 10 × 10 × 10 × 10 × 10 = 106. Now I can try a really big number like a sextillion. Since a sextillion has 21 zeros, we can write it as 1021. Then the number of cups in the ocean, thirteen sextillion is
13 × 1021 = 1.3 × 1022
The last way of writing it is called the normalized form.
What about numbers that are not so simple as twelve and a sextillion, such as the speed of light?
Not only is scientific notation compact, but it can also tell is something about the character of a number. 2.997 in the above example tells us that we know this number to four digits of accuracy. Actually this is a bad example, because as we will see later, we really know this number very well.
For all of its usefulness, scientific notation is a relatively new way of writing down a number. It dates back only to the 1950s and appeared first in computer manuals. In computer notation the number of cups in the ocean would look like 1.3E+22 and the speed of light would be written as 2.997E+08. If you looked at scientific papers a century ago all the digits of a numbers were usually written out, but the people building the first computers wanted a simple way of storing a number in memory that was independent of how big the number was. It is also easy to do things like multiplying numbers represented by scientific notation. For example, we can calculate one of the numbers mentioned at the beginning of this chapter, the number of meters in a light-year. The number of seconds in a year is about 3.15 × 107, and so we can combine this with the speed of light to get;
What made this simple was that I did not multiply seven-digit numbers by eight-digit numbers. Instead I multiplied 3.15 × 2.99 = 9.42 and added the exponents: 7 + 8 = 15.
So now we have a notation with which we can express really big numbers. Let us try to answer the question, “How many molecules are in the ocean?” To do this we are going to need something called Avogadro’s number, and when we write down this number we will really need scientific notation.
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Amedeo Avogadro (1776–1856) did not know how many molecules there are in a glass of wine, but if asked he would probably tell us that the number was very large but not infinite. In the development of the modern atomic and molecular description of matter Avogadro comes near the beginning, but not quite at the advent. That distinction goes to John Dalton (1766–1844), an English scientist whose interest lay not only in chemistry, but also in meteorology and color blindness, a condition with which he was afflicted. He was born in Cumbria, in northern England, he eventually moved to the bustling industrial city of Manchester where he became secretary of the Manchester Literary and Philosophical Society. It was at their meetings and in their publications where Dalton’s atomic theory unfolded. Much of Dalton’s theory remains intact even today. Matter is made of atoms, atoms come in different types, one type for each element, and atoms combine in unique ways to make molecules. It may seem obvious to us today, but Dalton was unique in that he was able to look at the chemical data of his day and see that atoms are real.
One of the key piece of evidence for Dalton is something called the law of definite proportions. For example, if I mix one part oxygen with two parts hydrogen it would all combine into water vapor. If I had used more hydrogen I would get the same amount of water vapor, as well as some residual hydrogen that did not react. It is the fact that there is this left-over residual that tells us about the atomic nature of matter.
If the law of definite proportions also applied to baking it would mean that when I combined sugar and flour to make cookies there would be only one correct ratio of the ingredients. If I had put in too much sugar, after I took the cookie dough out of the bowl, the extra sugar would still be laying there. This does not happen because cooking is generally about
mixing and not about chemistry. Too much sugar and I just get a very sweet cookie.
Back to chemistry. When I say “one part oxygen with two parts hydrogen,” what do I mean by the word “part”? When I am doling out the oxygen am I to measure its mass or volume, or something else?
Dalton said that for each element each part had a certain mass or weight. So for example, one part of hydrogen would be 1 g, whereas one part of oxygen was 8 g. With this technique he could explain a great number of reactions, but not everything. In fact it is this atomic weight that distinguishes his theory from that of the Greeks.
Most of the best chemical data of that time came from Joseph Louis Gay-Lussac (1778–1850), a professor at the Sorbonne, in Paris. Gay-Lussac studied various properties of gases, and even rode a hot air balloon up 6 km to study the chemistry of the atmosphere. But it was his data in the hands of Dalton and Avogadro that has had a lasting impact. Avogadro, after reading Gay-Lussac’s Memoir, proposed that “part” meant volume and, although we rarely think of it in those terms today, he was in fact right.
So how is it that Dalton and Avogadro, two bright and gifted scientists, can both look at Gay-Lussac’s data and come to very different conclusions? Their debate spanned a wide range of reactions that Gay-Lussac had measured, but we can understand the crux of the problem by looking at plain and simple water. Gay-Lussac reported that 2 l of hydrogen and 1 l of oxygen yielded 2 l of water vapor. So here is the problem from John Dalton’s point of view. The mass of 2 l of hydrogen is 0.18 g, and 1 l of oxygen is 1.4 g. Their ratio is about 1:8, just as Dalton’s atomic weights said. So according to Dalton, there are the same number of hydrogen molecules as oxygen molecules in these two samples.
Alternatively, according to Avogadro’s hypothesis, “the number of integral molecules in any gas is always the same for equal volumes.” So (according to Dalton) 2 hydrogen plus 1 oxygen should lead by Avogadro’s hypothesis to 1 l of water vapor. But instead it leads to 2 l. Dalton saw this as evidence to support his view. Still, Avogadro maintained that the distance between molecules is the same in all gases, and so the number of molecules in a liter was constant.
We now know that Avogadro was right: the distance between molecules of any gas, at the same temperature and pressure, is a constant. Given a liter of oxygen, or hydrogen, or water vapor, or ammonia, or anything else there will be the same number of molecules in each and every liter.
But how do we reconcile that with Dalton’s objections and Gay-Lussac’s data? The crux of the problem is that Avogadro talked about molecules and Dalton talked about atoms. If I have a liter bottle full of hydrogen, it is molecular hydrogen, or H2. Molecular hydrogen means that two hydrogen atoms are bound together. In fact they are bound close to each other and as far as their gas properties are concerned they act as a single object. They travel, bounce, mix and diffuse as one. They determine pressure and temperature and are the objects that Avogadro was talking about. So if we look at Gay-Lussac’s data, we now understand that 1 l of hydrogen contains twice as many atoms as molecules. This is true of oxygen as well, but not water. Water is singular.
All the reactions in Gay-Lussac’s data make sense if we understand that some gases (not all) naturally occur as two-atom molecules. So what about Dalton’s view of the data? Dalton assumed that simple gases (hydrogen, oxygen and so forth) were always atomic. So he explained the data by seeing water as HO! In the end, for a given volume of gas there is a set number of molecules, independent of the gas type. That is Avogadro’s hypothesis, but not Avogadro’s number. We still do not know how many molecules there are in a liter of a gas, or a gram of a solid, or a cup of the ocean. Loschmidt would be the first to measure the size of a molecule, half a century after Avogadro and Dalton, but to do that he needed to know something about the way molecules bounced around in a gas.
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The study of gases was a hot topic throughout the nineteenth century, but it was nearly 50 years later before we hit the next critical step in figuring out how many molecules there are in a cup of wine. James Clark Maxwell (1831–1879), who later would become famous for his unification of electricity, magnetism and light, was studying the diffusion and mixing of gases. He was working in the 1860s and theories and experiment were in disagreement. So he developed a new model for the distribution of motion in a gas, the kinetic theory of gases. In his description, a gas is a collection of molecules that are in motion, moving around and colliding with each other. The hotter the gas, the more they move. The more they collide, the higher the pressure. At one point Maxwell wrote
If we suppose feet per second for air at 60°, and therefore the mean velocity v = 1505 feet per second, then the value l, the mean distance traveled by a particle between consecutive collisions of an inch, and each particle makes 8,077,200,000 collisions per second.
I have included this whole paragraph because of the way the numbers are reported is interesting. First the 8 billion (short scale) or 8 thousand million (long scale) collisions per second is reported with all of its digits. Maxwell also seems to indicate five digits of accuracy, a precision that the raw data did not justify. He also reports the distance as a fraction, instead of the decimal form we presently would expect. But really it is this distance, the distance molecules travel between collisions, m, that will help us solve the wine glass problem.
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The next step came from Johann Josef Loschmidt (1821–1895), a scientist from Vienna who used Maxwell’s results to give us our first really good estimate of the size of a molecule and the magnitude of Avogadro’s number. Loschmidt had read all about the kinetic theory of gasses and he knew all about Maxwell’s results and that the distance that a molecule travels between collisions is of an inch. With this Loschmidt set out to calculate the number of molecules in a given volume of gas.
Loschmidt’s reasoning went something like this. The amount of elbowroom that a molecule needs is the volume of a cylinder whose cross sectional area is the same as the cross-section of the molecule, and whose length is the distance a molecule travels before it hits something (see Figure 3.1). If there was more space than this, then a molecule would travel further before it collided. The next thing that Loschmidt needed to know was how much of a gas volume is occupied by molecules, and how much is empty. He called this the condensation factor, and measured it by looking at the difference in volume between a gram of liquid water and water vapor. Typically the condensation factor is about a thousand. From that he showed that the size of a molecule is the ratio of the collision distance and the condensation factor. In other words, since the condensation factor told us that only one part in a thousand of a gas is not void, then the size of the molecule is about one thousandth of the collision length. Now actually Loschmidt’s equation is a bit more complex because he treats a molecule as a sphere and corrects for how tightly they can be packed to make a solid or liquid, but those are small corrections. In the end he reports that the size of a molecule is “0.000 000 969 mm” (9.69 × 10−10 m). He was measuring air, which is dominated by N2 (nitrogen), and so by modern measurements was only off by a factor of three. But he had the right magnitude. Also it was not his reasoning, but rather the experimental numbers that were supplied to him that account for most of the differences. He described his results as follows.
Figure 3.1 Loschmidt’s method for determining the size of a molecule. (A) This cylinder represents the average space a molecule occupies, where r is the radius of the cylinder and the molecule, and L is the distance traveled between collisions. (B) When the gas is condensed many molecules fit in the same volume. We can see that L = 2rC, where C is the condensation factor. From this, r, the size of the molecule, can be determined.
An imposing string of numbers such as our calculations yield, especially when taken into three dimensions, means that it is not too much to say that they are the true residue of the expectations created when microscopists have stood at the edge of the bottomless precipice and described them as “infinitesimally small.”
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I think Loschmidt must have been a bit awed by his results. It is not too often that a scientist will wax so poetically, especially when writing in a journal for his colleagues to read. But he really did have a result that was breathtaking. When he stood at the edge of that precipice, it was not that he saw that it had an infinite or unknown bottom, but rather that he was the first to get a glimpse of the bottom and to know that it was down there, nine or ten orders of magnitude smaller than a meter.
Loschmidt was so fascinated by this result that he spent part of that 1865 paper trying to give the readers a feel for what this small size meant. He explained that if all the atoms in a cubic millimeter were spread out to one layer thick, they would cover 1 m2 (actually 10 m2). He also reported that the wavelength of light is about 500 times longer than these molecules (actually about 1500 times). He also talked in terms of the weight of an atom:
Incidentally, the calculation suggests for the “atomic weight” of chemists a suitable unit of a trillionth of a milligram.
This suggestion of Loschmidt was interesting for three reasons. First off, he was suggesting something like what we now call an atomic mass unit (u). Today we define an atomic mass unit as one twelfth of the mass of a carbon-12 atom, which is about 1.66 × 10−21 mg. The second point of interest was that Loschmidt proposed a unit that was strictly metric. His number was 1.00 × 10−21 mg. It is curious that he would suggest that the basic unit be defined in terms of the macroscopic gram, instead of the microscopic atom. The choice of an atomic mass unit in terms of carbon ended up being a practical solution. It is easier to measure the ratio of the mass of a molecule to a carbon atom than the mass of that molecule to a macroscopic gram.
How Big is Big and How Small is Small Page 5