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Counting Molecules Using Avogradro's Number

To look at gases on the molecular level, you need to know how many molecules you have in a certain sample. Physicists use a measurement called a mole to relate the mass of a sample to the number of molecules it contains, and they use Avogadro’s number to represent the number of atoms in one mole.

A mole (abbreviated mol) is the number of atoms in 12.0 grams of carbon isotope 12. Carbon isotope 12 — also called carbon-12, or just carbon 12 — 
is the most common version of carbon. Isotopes of an element have the same number of protons but different numbers of neutrons. Carbon-12 has six protons and six neutrons (a total of 12 particles in its nucleus); however, some carbon atoms (isotopes) have a few more neutrons in them — carbon-13, for example, has seven neutrons. The average mass of a mole of a mixture of the carbon isotopes found in nature works out to be 12.011 grams.

The number of atoms in one mole (in 12.0 grams of carbon-12) has been measured as

image0.png

which is called Avogadro’s number, NA.

Do you find the same number of atoms in, say, 12.0 grams of sulfur? Nope. Each sulfur atom has a different mass from each carbon atom, so even if you have the same number of grams, you have a different number of atoms.

How much more mass does an atom of sulfur have than an atom of carbon-12? If you check the periodic table of elements hanging on the wall in a science lab, you find that the atomic mass of sulfur is 32.06. (Note: The atomic mass usually appears under the element’s symbol.) But 32.06 what? It’s 32.06 atomic mass units, u, where each atomic mass unit is 1/12 of the mass of a carbon-12 atom.

image1.png

12.0 grams, and the mass of your average sulfur atom is bigger than the mass of a carbon-12 atom:

Sulfur mass = 32.06 u

Carbon 12 mass = 12 u

Therefore, a mole of sulfur atoms must have this mass:

image2.png

How convenient! A mole of an element has the same mass in grams as its atomic mass in atomic units. You can read the atomic mass of any element in atomic units off any periodic table. For instance, you can find that a mole of silicon (atomic mass: 28.09 u) has a mass of 28.09 grams, a mole of sodium (atomic mass: 22.99 u) has a mass of 22.99 grams, and so on. Each of those moles of different substances contains

image3.png

Now you can determine the number of atoms in a diamond, which is solid carbon (atomic mass: 12.01 u). A mole is 12.01 grams of diamond, so when you find out how many moles you have, you multiply that times

image4.png

Then if you like, you can work out how many atoms of carbon are in a 1 carat diamond: 1 carat is equal to 0.200 grams, so here’s how many atoms you have:

image5.png

Not every object is made up of a single kind of atom. When atoms combine, you have molecules. For example, water is made up of two hydrogen atoms for every one oxygen atom (H2O). Instead of the atomic mass, you look for the molecular mass, which is also measured in atomic mass units. For example, the molecular mass of water is 18.0153 atomic mass units, so 1 mole of water molecules has a mass of 18.0153 grams.

Some physics problems provide the molecular mass; others require you to calculate molecular mass by using the atomic mass and the compound’s molecular formula. That is, you add the atomic masses of the individual atoms in the molecule.

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