The mole is a bridge between our world and the atomic world 

                                       Mole Day is celebrated among chemists on 
                                       October 23, between 6:02 a.m. and 6:02 p.m.
                                       The time and date derived from Avogadro's
                                       number, which is 6.02 X 1023, defining
                                       the number of molecules in a mole, one of  
                                       seven basic SI units.

                                       Your Encyclopedia

Bulk matter and the mole

We live in the macroscopic world. Exploring that world through our senses alone limits the scales of length, time, and mass that we can examine and measure directly. The range of scales of sizes of objects in the physical world varies enormously.

Specific scientific disciplines typically focus on the study of a limited range of length scales of objects. For example, the focus of astronomy is on the very large, while elementary particle physics focuses on the very small. Chemistry focuses on the study of the properties of atoms and molecules. Atoms are typically only a few ten-billionths of a meter in size. The astounding smallness of atoms is the reason that it took mankind a million years to propose that atoms even exist, and a few thousand years to prove it. The existence of atoms has been widely accepted for less than two hundred years. That is true in spite of the fact that everything around us is made of atoms!

Consider the following table of length scales. Can you recognize the range of scales studied by geography? How about biology? And chemistry? 

                         Length (m)       Object
                         1 X 1026          To Edge of Universe
                         3 X 1023          To Nearest Galaxy
                         3 X 1020          To Center of Milky Way
                         1 X 1017          To Nearest Star
                         1 X 1011          To Sun
                         4 X 108           To Moon
                         4 X 107           Around the Earth
                         5 X 106           Across the US
                         2 X 104           City
                            2              You
                         1 X 10-4          Salt grain
                         1 X 10-5          Yeast
                         1 X 10-6          Bacterium
                         1 X 10-7          Cell nucleus
                         4 X 10-8          Virus
                         1 X 10-8          DNA
                         3 X 10-9          Protein
                         1 X 10-9          β-carotene
                         1 X 10-10         Atom diameter
                         1 X 10-15         Nucleus diameter

We now know that atoms exist, but their sizes are terribly small

atomic diameter = 1-5 Å

and their masses are correspondingly small

atomic mass = 2 - 400 X 10-27 kg

Working in a chemical laboratory, it is convenient to measure chemicals on a gram scale. That way, the amounts are not so small that we lose them, and not so large that we can't pick them up!

If we want enough atoms to weigh accurately on a nineteenth century balance, we will need lots of them. Think of one gram of hydrogen atoms. Not much hydrogen, you might say. How many atoms is in that one gram of hydrogen?

# H atoms in one gram = (1.0 g / 1.67 X 10-27 kg/atom) X (1.0 kg / 103 g) = 6.0 X 1023 atoms

That's a lot of hydrogen atoms!

Now a baker often sells more than one bagel. He often will sell about twelve bagels at a time. So the baker uses a special word to describe that convenient unit of twelve -- he calls it a dozen. In chemistry we have the same situation. The nineteenth century chemist wanted to study a few grams of material, something that could be seen and weighed and poured. That few grams of atoms will usually have about 1024 atoms or molecules in it. So that convenient unit we call a mole. The chemist's mole is like the baker's dozen -- it is a most convenient number of things.

What exactly should the mole be? It is a bit arbitrary. The number of atoms or molecules or bagels in a mole is not a fundamental physical constant. It is a number selected to be useful, but there is no best value.

Kary Mullis (pictured with his longboard) is the inventor of the polymerase chain reaction (PCR) and an extraordinary character. He has reflected on a wide variety of scientific matters, including the definition of Avogadro's number. As he put it "Avogadro's number is not an intrinsic chemical constant. It is derived from a measurement of the size of the Earth heroically made by the French in the eighteenth century. The number depends on the Gram which came from the Cubic Centimeter full of water at 4oC, which derived from the Meter, which had been defined as one ten-millionth of the distance from the North Pole to the Equator along a line passing between the Notre-Dame and the guy selling chestnuts there in the winter of 1799." [Mullis in "PCR Methods and Applications," Cold Spring Harbor Laboratory Press, NY (1991)].

It was in 1811 that Amedeo Avogadro had his brilliant insight that a fixed volume of gas, at a given temperature and pressure, will contain an equal number of atoms and molecules, regardless of its composition. So at a given temperature and pressure, one cubic centimeter of hydrogen gas will contain the same number of molecules as one cubic centimeter of oxygen gas. The extraordinary reason that is true was not understood for many yeas. It would not be until 1865 that the absolute number of molecules in a cubic centimeter of air was estimated by Josef Loschmit to be 2.6 X 1019 molecules (under a standard temperature and pressure). It was Avogadro that had the insight, but Loschmit first estimated the number. That is why some people refer to Loschmit's number, rather than Avogadro's number.

At first, the mole was defined to be the number of hydrogen atoms in one gram of 1H. Then it was defined to be the number of oxygen atoms in 16 grams of 16O. So what is the current definition of the mole? It happens that this is the Age of Biology and carbon is the ``atom of life.'' It is organic, carbon-based molecules that are fundamental to all living organisms. So the mole is defined in honor of carbon to be

# C atoms in twelve grams 12 C = 6.022 137 X 1023 atoms/mole = 1 mole of atoms

Remember that this is a convenient constant. And while we can't write out all the digits, it is a counting number, just like a dozen.

As a result of the arbitrary nature of Avogadro's number, Mullis has argued that concentrations should be measured in "number of things per milliliter" instead of "moles per milliliter." That makes sense! And there is good reason to believe that idea could gain support. For while Avogadro (pictured here sans longboard) was a genius of the first order, there is little doubt that he was not remotely as cool as Mullis.

The existence of Avogadro's number can also be used to discount "scientific" ideas that are a bit too interesting. The fact that Avogadro's number is a counting number - a number of things - implies that a finite number of atoms exist in a given amount of material. That is the basis for scientific criticism of homeopathy, in which medicinal materials are so thoroughly diluted that Avogadro's number would imply that not a molecule of the medicinal material remains!

We can use the mole to count anything we like. We can talk about a mole of bagels, a mole of moles, a mole of bicycles, a mole of dollars, or a mole of stars. However, it turns out that the unit of the mole is really only useful in chemistry to count atoms and molecules.

Atomic size

How did people learn about the absolute sizes of atoms and molecules? Here is one way to do it. You begin with an observation concerning the nature of gases, liquids, and solids. Gases are compressible while liquids and solids aren't. You can put a gas in a piston, or a syringe, and you can change its volume quite a bit by squeezing it. However, you can't easily change the volume of a liquid or solid. If that doesn't seem right to you, give it a try.

From that observation, we can make the hypothesis that

In solids and liquids the atoms and molecules are closely packed.

So in gases, we imagine the atoms and molecules roaming around in a largely empty box. We can see that from the difference in density. If we take a volume of liquid water and vaporize it, we find that the liquid water is 1,000 times more dense than the gaseous water. And that is the case for most gases and liquids. On the other hand, when you melt a solid to form a liquid, there is usually very little change in the density.

Let's think about water. Since the molecules in liquid water are ``touching'' each other, we can figure out how large they are with very little effort given three bits of information: the density of liquid water (1.0 g/cm3), the molar mass of water (18.0 g/mole), and Avogadro's number (6.022 X 1023 molecules/mole). Think for a minute about how we can combine these three numbers so that the dimensions, or units, tend to cancel out. The most obvious way is

18.0 g/mole / 6.022 / 1023 molecules/mole X 1.0 g/cm3 = 30 X 10-24 cm3/molecule

What is that number exactly? It is the volume of a single water molecule! From that we can determine the water molecule's diameter by assuming that each water molecule is a small cube of volume Vcube = d3 so that

dmolecule = (30 X 10-24 cm3)1/3 = 3.1 X 10-8 cm = 3 Å

How were we able to accomplish measurement on the scale of atoms and molecules? Through the use of Avogadro's number which is a scale factor relating the size of a collection of atoms and molecules that we can hold in our hand to the the size of a single atom or molecule. But hold on! How do we determine Avogadro's number?