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# Lesson 3.4: Avogadro's Number and the Mole

← Lesson 3.3: Lesson 3.5: →
• To analyze the transformations that occur between individual atoms or molecules in a chemical reaction, we need to know how many atoms or molecules are contained in a measurable quantity in the laboratory—a given mass of sample.
• However, even in the smallest measurable mass there are enormous numbers of atoms.
• To manage these huge numbers, chemists deal with large collections of atoms of a known mass.
• The unit for that “large collection” is the mole (mol), from the Latin moles, meaning “pile” or “heap.”
• Many familiar items are sold in numerical quantities with distinct names.
• For example, eggs are sold by the dozen (12).
• Sheets of printer paper are packaged in reams of 500.
• In the same way a mole is 6.02 x 1023.
• It feels more confusing because the number is so huge and because it feels somewhat arbitrary. Why this and not some other number?

Why such a strange number?

If a scale could be invented to read in amu, one atom of H would read 1 amu and one atom of carbon would read 12 amu.

In the same way 2 atoms of H would read 2 amu and 2 atoms of carbon would read 24 amu. In general we can say that the mass on the carbon scale will always be 12 times that on the hydrogen scale when the number of atoms is the same.

Now switch to a real scale that reads in grams. The scale on the right contains a huge number of atoms! We can think of it as a "box" full of carbon atoms. Since the scale on the left has 12 times less mass of hydrogen it has the same huge number of atoms. The numberis 6.02 x 1023. The scale on the left has 1 mole or 6.02 x 1023 hydrogen atoms - a "box" full of hydrogens. The scale on the right has 1 mole or 6.02 x 1023 carbon atoms - a "box" full of carbons.

• Why such a strange number?
• A mole is defined as the amount of a substance that contains the number of carbon atoms in exactly 12 g of isotopically pure carbon-12.
• The number seems arbitrary because it is based on the familiar unit gram which has no inherent relationship to numbers of atoms.
• It is a raw fact that 12.000 g of carbon-12 contains 6.02 x 1023 atoms.
• And since we know the relative mass of all atoms, the relative mass of any atom (known in amu) is the mass of 1 mole of that atom when expressed in grams.
• This number is called Avogadro’s number, after the 19th-century Italian scientist who first proposed a relationship between the volumes of gases and the numbers of particles they contain.

Here are samples of 1 mole, or 6.02 x 1023 atoms, of different elements. Note that the number is just the relative mass of the atoms in the unit grams.

The molar mass of a molecule.

We can use the unit mole to count objects other than atoms. For example, we can count molecules in lots of moles. One mole of water would contain 6.02 x 1023 molecules of H2O. It would contain 2 moles of hydrogen and 1 mole of oxygen, and have a mass of 18.0 g. Note that the numerical value for the molar mass is same as the mass of a single molecule, 18.0 amu, in the unit grams - 18.0 g.

We find the molar mass (the mass of one mole) of a molecule of a substance in the exact same way that we found the fomula weights in Lesson 3.3, but express it in the unit grams.

## Example 1: How many moles of water are in a 36.0 g sample of water?

The molar mass of water is 18.0 g, so we have two moles of water in 36.0 g.

We can set up a ratio. There are x moles in 36.0 g when for every 1.0 mol there are 18.0 g.

$$\frac {x}{36.0 \, g} = \frac{1 \, mol}{ 18.0 \, g}$$

Solving for x we get

\begin{align*} x &= 36.0 \, g \times \frac{1 \, mol}{ 18.0 \, g} \\ x &= 2.0 \, mol \end{align*}

Note that we always do the same thing when looking for the number of moles of a substance: Divide by the molar mass.

## Example 2: How many moles of ammonia, NH3, are in a 340.0 g sample ?

The molar mass of ammonia is 17.0 g, so we have twenty moles of ammonia in 340.0 g.

We can set up a ratio. There are x moles in 340.0 g when for every 1.0 mol there are 17.0 g.

$$\frac {x}{340.0 \, g} = \frac{1 \, mol}{ 17.0 \, g}$$

Solving for x we get

\begin{align*} x &= 340.0 \, g \times \frac{1 \, mol}{ 17.0 \, g} \\ x &= 20.0 \, mol \end{align*}

Note that we always do the same thing when looking for the number of moles of a substance: Divide by the molar mass.

## Example 3: How many grams of water are in 6.0 moles ?

The molar mass of water is 18.0 g, so we have 6 x 18 = 108 grams of water in 6.0 moles.

We can set up a ratio. There are x grams in 6.0 mol when for every 18.0 g we have 1.0 mol.

$$\frac {x}{6.0 \, mol} = \frac{18.0 \, g}{ 1.0 \, mol}$$

Solving for x we get

\begin{align*} x &= 6.0 \, mol \times \frac{18.0 \, g}{ 1.0 \, mol} \\ x &= 108 \, g \end{align*}

Note that we always do the same thing when looking for the mass of a given number of moles of a substance: Multiply by the molar mass.

## Example 4: How many grams of ammonia are in 12.0 moles ?

The molar mass of ammonia is 17.0 g, so we have 17 x 12 = 204 grams of ammonia in 12.0 moles.

$$\frac {x}{12.0 \, mol} = \frac{17.0 \, g}{ 1.0 \, mol}$$