5.4: The Mole

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Skills to Develop

Use Avogadro's number to convert to moles and vice versa given the number of particles of a substance.
Use the molar mass to convert to grams and vice versa given the number of moles of a substance

Introduction

When objects are very small, it is often inconvenient or inefficient, or even impossible to deal with the objects one at a time. For these reasons, we often deal with very small objects in groups, and have even invented names for various numbers of objects. The most common of these is "dozen" which refers to 12 objects. We frequently buy objects in groups of 12, like doughnuts or pencils. Even smaller objects such as straight pins or staples are usually sold in boxes of 144, or a dozen dozen. A group of 144 is called a "gross".

This problem of dealing with things that are too small to operate with as single items also occurs in chemistry. Atoms and molecules are too small to see, let alone to count or measure. Chemists needed to select a group of atoms or molecules that would be convenient to operate with.

Avogadro's Number

In chemistry, it is impossible to deal with a single atom or molecule because we can't see them or count them or weigh them. Chemists have selected a number of particles with which to work that is convenient. Since molecules are extremely small, you may suspect this number is going to be very large and you are right. The number of particles in this group is \(6.02 \times 10^{23}\) particles and the name of this group is the mole (the abbreviation for mole is \(\text{mol}\)). One mole of any object is \(6.02 \times 10^{23}\) of those objects. There is a very particular reason that this number was chosen and we hope to make that reason clear to you.

When chemists are carrying out chemical reactions, it is important that the relationship between the numbers of particles of each reactant is known. Chemists looked at the atomic masses on the periodic table and understood that the mass ratio of one carbon atom to one sulfur was \(12 \: \text{amu}\) to \(32 \: \text{amu}\). They realized that if they massed out 12 grams of carbon and 32 grams of sulfur, they would have the same number of atoms of each element. They didn't know how many atoms were in each pile but they knew the number in each pile had to be the same. This is the same logic as knowing that if a basketball has twice the mass of a soccer ball and you massed out \(100 \: \text{lbs}\) of basketballs and \(50 \: \text{lbs}\) of soccer balls, you would have the same number of each ball. Many years later, when it became possible to count particles using electrochemical reactions, the number of atoms turned out to be \(6.02 \times 10^{23}\) particles. Eventually chemists decided to call that number of particles a mole. The number \(6.02 \times 10^{23}\) is called Avogadro's number. Avogadro, of course, had no hand in determining this number, rather it was named in honor of Avogadro.

Converting Between Molecules and Moles

We can use Avogadro's number as a conversion factor, or ratio, in dimensional analysis problems. If we are given a number of molecules of a substance, we can convert it into moles by dividing by Avogadro's number and vice versa.

Example 5.4.1

How many moles are present in 1 billion (\(1 \times 10^9\)) molecules of water?

Solution:

\[1 \times 10^9 \: \cancel{\text{molecules} \: \ce{H_2O}} \cdot \frac{1 \: \text{mol} \: \ce{H_2O}}{6.02 \times 10^{23} \: \cancel{\text{molecules} \: \ce{H_2O}}} = 1.7 \times 10^{-15} \: \text{mol} \: \ce{H_2O}\]

You should note that this amount of water is too small for even our most delicate balances to determine the mass. A very large number of molecules must be present before the mass is large enough to detect with our balances.

Example 5.4.2

How many molecules are present in \(0.00100 \: \text{mol}\)?

Solution:

\[0.00100 \: \cancel{\text{mol}} \cdot \frac{6.02 \times 10^{23} \: \text{molecules}}{1 \: \cancel{\text{mol}}} = 6.02 \times 10^{20} \: \text{molecules}\]

Converting Grams to Moles and Vice Versa

\(1.00 \: \text{mol}\) of carbon-12 atoms has a mass of \(12.0 \: \text{g}\) and contains \(6.02 \times 10^{23}\) atoms. Likewise, \(1.00 \: \text{mol}\) of water has a mass of 18.0 grams and contains \(6.02 \times 10^{23}\) molecules. 1.00 mole of any element or compound has a mass equal to its molecular mass in grams and contains \(6.02 \times 10^{23}\) particles. The mass, in grams, of 1 mole of particles of a substance is now called the molar mass (mass of 1.00 mole).

To quickly find the molar mass of a substance, you need to look up the masses on the periodic table and add them together. For example, water has the formula \(\ce{H_2O}\). Hydrogen has a mass of \(1.0084 \: \text{g/mol}\) (see periodic table) and oxygen has a mass of \(15.9994 \: \text{g/mol}\). The molar mass of \(\ce{H_2O} = 2 \left( 1.0084 \: \text{g/mol} \right) + 15.9994 \: \text{g/mol} = 18.0162 \: \text{g/mol}\). This means that 1 mole of water has a mass of 18.0162 grams.

We can also convert back and forth between grams of substance and moles. The conversion facto for this is the molar mass of the substance. The molar mass is the ratio giving the number of grams for each one mole of the substance. This ratio is easily found by adding up the atomic masses of the elements within a compound using the periodic table. This ratio has units of grams per mole or \(\text{g/mol}\).

Example 5.4.3

Find the molar mass of each of the following:

a) \(\ce{S}\)

b) \(\ce{H_2O}\)

c) \(\ce{F_2}\)

d) \(\ce{H_2SO_4}\)

e) \(\ce{Al_2(SO_4)3}\)

Solution:

a) Look for sulfur on the periodic table. Its molar mass is \(32.065 \: \text{g/mol}\). That means that one mole of sulfur has a mass of 32.065 grams.

b) This compound contains two hydrogen atoms and one oxygen atom. To find the molar mass of \(\ce{H_2O}\), we need to add the mass of two hydrogen atoms plus the mass of one oxygen atom. We get: \(2 \left( 1.008 \right) + 16.00 = 18.016 \: \text{g/mol}\). That means that one mole of water has a mass of just over 18 grams.

c) This compound contains two fluorine atoms. To find the molar mass of \(\ce{F_2}\), we need to add the mass of two fluorine atoms. We get: \(2 \left( 19.00 \right) = 38.00 \: \text{g/mol}\)

d) This compound contains two hydrogen atoms, one sulfur atom, and four oxygen atoms. To find the molar mass of \(\ce{H_2SO_4}\), we need to add the mass of two hydrogen atoms plus the mass of one sulfur atom plus the mass of four oxygen atoms. We get: \(2 \left( 1.008 \right) + 32.065 + 4 \left( 16.00 \right) = 100.097 \: \text{g/mol}\)

e) This compound contains two aluminum atoms, three sulfur atoms, and twelve oxygen atoms. To find the molar mass of \(\ce{Al_2(SO_4)_3}\), we need to add the mass of all of these atoms. We get: \(2 \left( 26.98 \right) + 3 \left( 32.065 \right) + 12 \left( 16.00 \right) = 342.155 \: \text{g/mol}\)

To convert the grams of a substance into moles, we use the ratio molar mass. We divide by the molar mass and to convert the moles of a substance into grams, we multiply by the molar mass.

Example 5.4.4

How many moles are present in 108 grams of water?

Solution:

\[108 \: \cancel{\text{g} \: \ce{H_2O}} \cdot \frac{1 \: \text{mol} \: \ce{H_2O}}{18.02 \: \cancel{\text{g} \: \ce{H_2O}}} = 5.99 \: \text{mol} \: \ce{H_2O}\]

To get the ratio \(1 \: \text{mol} \: \ce{H_2O} = 18.02 \: \text{g}\), we added up the molar mass of \(\ce{H_2O}\) using the masses on a periodic table.

Example 5.4.5

What is the mass of \(7.50 \: \text{mol}\) of \(\ce{CaO}\)?

Solution:

\[7.50 \: \cancel{\text{mol} \: \ce{CaO}} \cdot \frac{56.0 \: \text{g} \: \ce{CaO}}{1 \: \cancel{\text{mol} \: \ce{CaO}}} = 420 \: \text{g} \: \ce{CaO}\]

To get the ratio \(1 \: \text{mol} \: \ce{CaO} = 56.0 \: \text{g}\), we added up the molar mass of \(\ce{CaO}\) using the masses on a periodic table.

We will be using these ratio again to solve more complex problems in the next chapters. Being able to use these ratios is a very important skill for later math problems.

Lesson Summary

There are \(6.02 \times 10^{23}\) particles in 1.00 mole. This number is called Avogadro's number.
Th molar mass of a substance can be found by adding up the masses on a periodic table.
Using the factor-label method, it is possible to convert between grams, moles, and the number of atoms or molecules.

Vocabulary

Avogadro's number: The number of objects in a mole; equal to \(6.02 \times 10^{23}\).
Mole: An Avogadro's number of objects.
Molar mass: The mass, in grams, of 1 mole of a substance. This can be found by adding up the masses on the periodic table.

Introduction

Avogadro's Number

Converting Between Molecules and Moles

Converting Grams to Moles and Vice Versa

Lesson Summary

Vocabulary

Further Reading/Supplemental Links

Contributors