I ask you to go and get me one marble. Well, that"s easy; you don"t even need to count. Who needs to count to one?

If I sent you for ten marbles, that would be easy, too. Even 100, though it would take longer to count.

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But what about 10 million marbles? How long would that take to count? Assuming you could count three marbles per second, that would be over 38 days of continuous counting—no sleep.

There"s got to be a better way or we just couldn"t get the job done. If you"re clever, and I think you are, you"ll do something like weigh a hundred marbles, then multiply that by 100,000 to get the approximate weight of ten million marbles (probably to within less than a percent error), then you could just weigh out the marbles I asked for pretty quickly.

That"s the principle behind the mole in chemistry. It"s a bridge between the number of things and the mass of some known amount of things. It"s that simple.

### Why we need moles – an example

Take a look at this simple synthesis reaction, in which two hydrogen molecules (H2) combine with one oxygen molecule (O2) to produce two molecules of water: Now suppose we want to run this reaction, but run it in such a way that we mix together just the right amount of each reactant so that at the end of the reaction there"s no extra H2 or O2 left over, just H2O.

The balanced equation says that we need to have two H2 molecules for every O2. So we need to "count out" twice as much hydrogen as oxygen. But how do we do that? These molecules are very small. VERY small.

The trick, again, is to have a nice connection, our bridge, between numbers of atoms/ molecules, and mass; we need to know how many atoms of a certain kind are in some given mass, but remember that the atoms of every element have a different mass.

### The mole

It"s a funny name, the "mole." It doesn"t have anything to do with the varmint that burrows under ground. Much of early chemistry was developed by German chemists, and the word "mole" is the English version of the German word "mol" which is short for molekulargewicht, or "molecular weight." So it"s not so odd after all.

We"ll discuss the particulars below, but the mole is basically a known relationship between the mass of a collection of atoms and the number of atoms in that collection. For historical reasons, the mole happens to be the number of atoms in exactly 12.0 grams of pure carbon, but we"ll get to that later.

The figure below shows how having the mole (this one is just made up: 12 particles has a mass of 10 g) can serve as the bridge between mass and number. If we know the mass of a known number of particles, we can divide by the mass per number (our "mole") and get the number of particles in that mass.

If we know the number of particles, we can multiply by the mass per number to get the mass. This ability, simple as it seems, will be invaluable in our study of chemistry. Notice that in the calculations above I"ve carefully written out and canceled the units to make sure that the calculation represents the conversion I really want to make. You should do that, too.

### It starts with carbon

We begin, for reasons tied to the historic development of chemistry, with carbon. If we measure the mass of one element in an instrument called a mass spectrometer, the result is meaningless because a mass spec. can only give us relative masses. That is, it can tell us how much heavier or lighter one element is than another – in multiples of the mass of a proton or neutron, but nothing absolute. We don"t have a scale for directly measuring the weight of atoms.

So early on, we made a decision: We set the mass of carbon to 12, in units we called atomic mass units (amus) because most carbon has six protons and six neutrons, and they constitute most of the mass of the atom. Then when we sent other elements through the mass spectrometer, we would get their masses in multiples or fractions of the carbon mass.

For example, Lithium (Li), would have half the mass of carbon (because it has half the number of heavy particles in its nucleus). Magnesium (Mg) has a mass twice that of carbon, and so on.

In this way, the relative masses of the elements were measured and the periodic table was ordered by mass (among other atomic properties). Much later, masses were adjusted using further knowledge, thus the 12.01115 amu mass for carbon in the figure above*.

*Different periodic tables are based upon rounding of one element to round figures, some use hydrogen, some oxygen and some carbon.

Now it"s not surprising that the mass of a group of atoms or molecules is directly proportional to the number of atoms or molecules present in the sample. Nor is it surprising that the mass of an atom is proportional to the number of heavy particles (protons and neutrons) in its nucleus.

If one carbon atom weighs 12 amu, then two will weigh 24 amu, and so on. We"d like to be able to measure the masses of elements like carbon in grams, because amu"s are very small units that we can"t actually weigh with ease.

So the mass of carbon in grams has to be proportional to the number of atoms present in the sample.

What if, for convenience, we made a number – our mole – be the number of atoms in 12 grams of carbon?

This number was in fact measured in several ways around 1910 by physicist Jean Perrin, and he named the special number "Avogadro"s number" after Amadeo Avogadro, who in about 1810 had proposed that the volume of a gas is proportional to the number of gas atoms present. Avogadro"s number (L) is about 6.022 x 1023 atoms per mole.

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Now, here"s the beauty of this number: Let"s think (just as an example) about Lithium (Li), which, with three protons and three neutrons in its nucleus, has half the atomic mass of carbon. The same number of atoms, each of which weighs half the mass of carbon, should produce a total mass of half of our 12 grams of carbon. That means that in 6 g of Li, there are 6.022 x 1023 Li atoms. It turns out that there are 6.022 x 1023 atoms of any element in n grams of that element, where n is its atomic mass. It"s a very special number.