# 8.02: The Mole

- Page ID
- 178152

Learning Objectives

- Describe the unit
*mole*. - Relate the mole quantity of substance to its mass.
- Relate atoms of an element to the mass of an element or moles of an element

So far, we have been talking about chemical substances in terms of individual atoms and molecules. Yet we do not typically deal with substances an atom or a molecule at a time; we work with millions, billions, and trillions of atoms and molecules at a time. What we need is a way to deal with macroscopic, rather than microscopic, amounts of matter. We need a unit of amount that relates quantities of substances on a scale that we can interact with.

Chemistry uses a unit called mole. A **mole **(mol) is a number of things equal to the number of atoms in exactly 12 g of carbon-12. Experimental measurements have determined that this number is very large:

1 mol = 6.02214179 × 10^{23} things

Understand that a mole means a number of things, just like a dozen means a certain number of things-twelve, in the case of a dozen. But a mole is a much larger number of things. These things can be atoms, or molecules, or eggs; however, in chemistry, we usually use the mole to refer to the amounts of atoms or molecules. Although the number of things in a mole is known to eight decimal places, it is usually fine to use only two or three decimal places in calculations. The numerical value of things in a mole is often called *Avogadro's number* (*N*_{A}), which is also known as the *Avogadro constant*, after Amadeo Avogadro, an Italian chemist who first proposed its importance.

Example \(\PageIndex{1}\):

How many molecules are present in 2.76 mol of H_{2}O? How many atoms is this?

**Solution**

The definition of a mole is an equality that can be used to construct a conversion factor. Also, because we know that there are three atoms in each molecule of H_{2}O, we can also determine the number of atoms in the sample.

\[2.76\, \cancel{mol\, H_{2}O}\times \frac{6.022\times 10^{23}molecules\, H_{2}O}{\cancel{mol\, H_{2}O}}=1.66\times 10^{24}molecules\, H_{2}O\]

To determine the total number of atoms, we have

\[1.66\times 10^{24}\cancel{molecules\, H_{2}O}\times \frac{3\, atoms}{1\, molecule}=4.99\times 10^{24}\, atoms\]

Exercise \(\PageIndex{1}\)

How many molecules are present in 4.61 × 10^{−2} mol of O_{2}?

**Answer**

2.78 × 10^{22} molecules

How big is a mole? It is very large. Suppose you had a mole of dollar bills that need to be counted. If everyone on earth (about 6 billion people) counted one bill per second, it would take about 3.2 million years to count all the bills. A mole of sand would fill a cube about 32 km on a side. A mole of pennies stacked on top of each other would have about the same diameter as our galaxy, the Milky Way. A mole is a lot of things-but atoms and molecules are very tiny. One mole of carbon atoms would make a cube that is 1.74 cm on a side, small enough to carry in your pocket.

Why is the mole unit so important? It represents the link between the microscopic and the macroscopic, especially in terms of mass. *A mole of a substance has the same mass in grams as one unit (atom or molecules) has in atomic mass units*. The mole unit allows us to express amounts of atoms and molecules in visible amounts that we can understand.

For example, we already know that, by definition, a mole of carbon has a mass of exactly 12 g. This means that exactly 12 g of C has 6.022 × 10^{23} atoms:

12 g C = 6.022 × 10^{23} atoms C

We can use this equality as a conversion factor between the number of atoms of carbon and the number of grams of carbon. How many grams are there, say, in 1.50 × 10^{25} atoms of carbon? This is a one-step conversion:

\[1.50\times 10^{25}\cancel{atoms\, C}\times \frac{12.0000\, g\, C}{6.022\times 10^{23}\cancel{atoms\, C}}=299\, g\, C\]

But it also goes beyond carbon. Previously we defined atomic and molecular masses as the number of atomic mass units per atom or molecule. Now we can do so in terms of grams. The atomic mass of an element is the number of grams in 1 mol of atoms of that element, while the molecular mass of a compound is the number of grams in 1 mol of molecules of that compound. Sometimes these masses are called **molar masses **to emphasize the fact that they are the mass for 1 mol of things. (The term *molar* is the adjective form of mole and has nothing to do with teeth.)

## Converting from mass to atoms/molecules or atoms/molecules to mass

In the example above the number of atoms of carbon is directly related to the mass of one mole of carbon. Another way to do this, which may be more transparent is we know we can string conversion factors together, recall converting from seconds to years for example. This also holds true for our mole to mass and mole to atom conversions.

Example \(\PageIndex{6}\)

Calculate the mass (in grams) or 5.45 x 10^{26} atoms of iron.

**Solution**

We have two conversion factors we can use:

1 mole of Fe = 6.022 x 10^{23} atoms Fe

1 mole Fe = 55.854 g Fe

Using our conversion factors we have:

\[5.45 \,x\, 10^{26} \,\cancel{atoms\, Fe}\times\left( \frac{1\, \cancel{mole\, Fe}}{6.022 \,x\, 10^{23}\, \cancel{atoms\, Fe}}\right)\times\left(\frac{55.854 \,g \,Fe}{1 \cancel{mole\, Fe}}\right)\]

## Summary

The mole is a key unit in chemistry. The molar mass of a substance, in grams, is numerically equal to one atom's or molecule's mass in atomic mass units. We can use Avogadro's number and the concept of molar mass to determine the mass of a number of atoms or the number of atoms in a certain mass of an element.