10.1: The Mole Concept
Lesson Objectives
 Identify three methods for measuring the amount of matter in a sample.
 Define the mole and its relationship to Avogadro’s number.
 Use Avogadro’s number to convert between moles and the number of representative particles of a substance.
 Relate the atomic mass of an element to its molar mass.
 Calculate the molar mass of a given compound.
Lesson Vocabulary
 Avogadro’s number
 formula mass
 molar mass
 mole
 representative particle
Check Your Understanding
Recalling Prior Knowledge
 What is a conversion factor? What is dimensional analysis?
 What is meant by the atomic mass of an element, and in what units are atomic masses expressed?
 How are the structures of molecular compounds and ionic compounds different?
Chemistry is a quantitative science. It is not enough to simply observe chemical reactions and describe what happens. Chemists always need to know how much. How many liters of carbon dioxide gas are going to be produced for every gallon of gasoline that is burned? How many kilograms of the industrial chemical sulfuric acid are produced in a typical year? How many kilograms of sulfur, oxygen, and water are required to manufacture that much sulfuric acid? These types of questions show the quantitative nature of chemistry and chemical reactions.
How Much Matter?
If you do any baking, you may have a set of canisters in your kitchen that hold large amounts of flour and sugar so that they are easy to access when preparing some cookies or a cake. Think about how much sugar is in one of those canisters. How would you measure it? There are actually multiple answers to this question. One way to measure the amount of sugar in the canister is to find its mass. Another would be to measure its volume. A third way, and a very time consuming one, would be to count all of the individual grains of sugar. The amount of matter can be measured in three basic ways: mass, volume, and number of particles. In order to fully understand and manipulate chemical reactions, chemists must be able to understand these three ways of measuring matter and the interconnections between them.
Names for Numbers
How many bananas is a dozen bananas (Figure below)? How many elephants is a dozen elephants? How many asteroids is a dozen asteroids? For all of these questions, the answer is obviously twelve. There is nothing that we need to know about the bananas or the elephants or the asteroids in order to answer the question. The term dozen is always used to refer to twelve of something; it is a name that is given to an amount. Other amounts are given special names as well. For example, a pair is always two, and a gross of something is a dozen dozens, which would be 144.
Bananas can be sold by mass or by count.
We can use a conversion factor and dimensional analysis to convert back and forth between the number of items and the name given to a certain number. For example, if you wanted to know how many bananas there are in 8 dozens, you could perform the following calculation:



\begin{align*}8 \ \text{dozen bananas} \times \frac{12 \ \text{bananas}}{1 \ \text{dozen bananas}}=96 \ \text{bananas}\end{align*}
8 dozen bananas×12 bananas1 dozen bananas=96 bananas

\begin{align*}8 \ \text{dozen bananas} \times \frac{12 \ \text{bananas}}{1 \ \text{dozen bananas}}=96 \ \text{bananas}\end{align*}

The conversion factor of 12 items = 1 dozen items is true regardless of the identity of the item. Alternatively, you could find out how many dozens of asteroids are in a collection of 1242 asteroids.



\begin{align*}1242 \ \text{asteroids} \times \frac{1 \ \text{dozen asteroids}}{12 \ \text{asteroids}}=103.5 \ \text{dozen asteroids}\end{align*}
1242 asteroids×1 dozen asteroids12 asteroids=103.5 dozen asteroids

\begin{align*}1242 \ \text{asteroids} \times \frac{1 \ \text{dozen asteroids}}{12 \ \text{asteroids}}=103.5 \ \text{dozen asteroids}\end{align*}

The conversion factor is simply inverted in this case so that the units cancel correctly.
Conversion factors can also be used to relate the amount of something to its mass. Suppose that you have a small bunch of bananas consisting of five bananas. You place them on a balance and find that the five bananas have a mass of 850 g. Assuming each banana has the same mass, what would be the mass of 3 dozen bananas? We can employ two conversion factors to find the answer.



\begin{align*}3 \ \text{dozen bananas} \times \frac{12 \ \text{bananas}}{1 \ \text{dozen bananas}} \times \frac{850 \ \text{g}}{5 \ \text{bananas}}= 6120 \ \text{g}= 6100 \ \text{g}\end{align*}
3 dozen bananas×12 bananas1 dozen bananas×850 g5 bananas=6120 g=6100 g

\begin{align*}3 \ \text{dozen bananas} \times \frac{12 \ \text{bananas}}{1 \ \text{dozen bananas}} \times \frac{850 \ \text{g}}{5 \ \text{bananas}}= 6120 \ \text{g}= 6100 \ \text{g}\end{align*}

By knowing the mass of five bananas, we now have a relationship that we can use to convert between mass and number of bananas for any number or any mass. Note that because 850 g was a measured quantity with two significant figures, the result was also rounded to two significant figures. The amounts 5 and 12 are found by counting, so they are exact quantities and have unlimited significant figures.
The Mole and Avogadro’s Number
It certainly is easy to count bananas or to count elephants (as long as you stay out of their way). However, you would be counting grains of sugar from your sugar canister for a long, long time. Recall from the chapter Atomic Structure that atoms and molecules are extremely small – far, far smaller than grains of sugar. Counting atoms or molecules is not only unwise, it is absolutely impossible. One drop of water contains about 10^{22} molecules of water. If you counted 10 molecules every second for 50 years without stopping, you would have counted only 1.6 × 10^{10} molecules. At that rate, it would take you over 30 trillion years to count the water molecules in one tiny drop. Wow.
Chemists needed a name that can stand for a very large number of items. Amedeo Avogadro (17761856), an Italian scientist (Figure below), provided just such a number. He is responsible for the counting unit of measure called the mole. A mole (mol) is the amount of a substance that contains 6.02 × 10^{23} representative particles of that substance. The mole is the SI unit for amount of a substance. Just like the dozen and the gross, it is a name that stands for a number. There are therefore 6.02 × 10^{23} water molecules in a mole of water molecules. There also would be 6.02 × 10^{23} bananas in a mole of bananas, if such a huge number of bananas ever existed.
Italian scientist Amedeo Avogadro, whose work led to the concept of the mole as a counting unit in chemistry.
The number 6.02 × 10^{23} is called Avogadro’s number, the number of representative particles in a mole. It is an experimentally determined number. A representative particle is the smallest unit in which a substance naturally exists. For the majority of pure elements, the representative particle is the atom. Samples of pure iron, carbon, and helium consist of individual iron atoms, carbon atoms, and helium atoms, respectively. Seven elements exist in nature as diatomic molecules (H_{2}, N_{2}, O_{2}, F_{2}, Cl_{2}, Br_{2}, and I_{2}), so the representative particle for these elements is the molecule. Likewise, all molecular compounds, such as H_{2}O and CO_{2}, exist as molecules, so the molecule is their representative particle as well. For ionic compounds such as NaCl and Ca(NO_{3})_{2}, the representative particle is the formula unit. A mole of any substance contains Avogadro’s number (6.02 × 10^{23}) of representative particles.
The animal mole (left) is very different than the counting unit of the mole. Chemists nonetheless have adopted the mole as their unofficial mascot (right). National Mole Day is a celebration of chemistry that occurs on October 23rd (10/23) of each year.
You can watch a video lecture about moles (the unit) by going to http://www.youtube.com/watch?v=AsqEkF7hcII.
Watch a video that shows an experimental calculation of Avogadro's Number http://www.youtube.com/watch?vp9QYJqFq5s.
Video experiment questions:
1. Briefly describe this experiment.
2. Write the calculations shown in this experiment.
3. What was the experimental result?
4. What was the percent error?
Conversions Between Moles and Number of Particles
Just as we did with dozens of bananas, we can use the number of items in a mole to convert back and forth between a number of particles and moles of those particles.
Sample Problem 10.1: Converting a Number of Particles to Moles
The element carbon exists in two primary forms: graphite and diamond. How many moles of carbon atoms are in a sample containing 4.72 × 10^{24} atoms of carbon?
Step 1: List the known quantities and plan the problem.
Known
 number of C atoms = 4.72 × 10^{24}
 1 mole = 6.02 × 10^{23}
Unknown
 4.72 × 10^{24} C atoms = ? mol C
One conversion factor will allow us to convert from the number of C atoms to moles of C atoms.
Step 2: Calculate



\begin{align*}4.72 \times 10^{24} \ \text{atoms C} \times \frac{1 \ \text{mol C}}{6.02 \times 10^{23} \ \text{atoms C}} = 7.84 \ \text{mol C}\end{align*}
4.72×1024 atoms C×1 mol C6.02×1023 atoms C=7.84 mol C

\begin{align*}4.72 \times 10^{24} \ \text{atoms C} \times \frac{1 \ \text{mol C}}{6.02 \times 10^{23} \ \text{atoms C}} = 7.84 \ \text{mol C}\end{align*}

Step 3: Think about your result.
The given number of carbon atoms was greater than Avogadro’s number, so the number of moles of C atoms is greater than 1 mole. Since our starting value is reported to three significant figures, the result of the calculation is also rounded to three significant figures.
 Convert the given number of particles to moles.
 3.65 × 10^{22} molecules of H_{2}O
 9.18 × 10^{23} formula units of KCl
 Convert the given number of moles to the number of representative particles.
 1.25 mol Zn atoms
 0.061 mol CH_{4} molecules
Suppose that you wanted to know how many hydrogen atoms were in a mole of water molecules. First, you would need to know the chemical formula for water, which is H_{2}O. There are two atoms of hydrogen in each molecule of water. How many atoms of hydrogen would there be in two water molecules? There would be 2 × 2 = 4 hydrogen atoms (Figure below). How about in a dozen? Since a dozen is 12, there would be 12 × 2 = 24 hydrogen atoms in a dozen water molecules. To get the answers, (4 and 24) you had to multiply the given number of molecules by two atoms of hydrogen per molecule. Similarly, to find the number of hydrogen atoms in a mole of water molecules, the problem could be solved using conversion factors.
\begin{align*}1 \ \text{mol H}_2\text{O} \times \frac{6.02 \times 10^{23} \ \text{molecules H}_2\text{O}}{1 \ \text{mol H}_2\text{O}} \times \frac{2 \ \text{atoms H}}{1 \ \text{molecule H}_2\text{O}} = 1.20 \times 10^{24} \ \text{atoms H}\end{align*}
The first conversion factor converts from moles of particles to the number of particles. The second conversion factor reflects the number of atoms contained within each molecule.
Two water molecules contain 4 hydrogen atoms and 2 oxygen atoms. A mole of water molecules contains 2 moles of hydrogen atoms and 1 mole of oxygen atoms.
Sample Problem 10.2: Atoms, Molecules, and Moles
Sulfuric acid has the chemical formula H_{2}SO_{4}. A certain quantity of sulfuric acid contains 4.89 × 10^{25} atoms of oxygen. How many moles of sulfuric acid are in the sample?
Step 1: List the known quantities and plan the problem.
Known
 number of O atoms in the sample = 4.89 × 10^{25}
 1 mol H_{2}SO_{4} = 6.02 × 10^{23} molecules H_{2}SO_{4}
Unknown
 mol of H_{2}SO_{4} molecules in the sample
Two conversion factors will be used. First, convert atoms of oxygen to molecules of sulfuric acid. Then, convert molecules of sulfuric acid to moles of sulfuric acid.
Step 2: Calculate
\begin{align*}4.89 \times 10^{25} \ \text{atoms O} \times \frac{1 \ \text{molecule H}_2\text{SO}_4}{4 \ \text{atoms O}} \times \frac{1 \ \text{mol H}_2\text{SO}_4}{6.02 \times 10^{23} \ \text{molecules H}_2\text{SO}_4} = 20.3 \ \text{mol H}_2\text{SO}_4\end{align*}
Step 3: Think about your result.
The original number of oxygen atoms was about 80 times larger than Avogadro’s number. Since each sulfuric acid molecule contains 4 oxygen atoms, there are about 20 moles of sulfuric acid molecules.
 How many atoms of carbon are in 0.750 moles of propane, which has a chemical formula of C_{3}H_{8}?
 The chemical formula of glucose is C_{6}H_{12}O_{6}. How many moles of glucose are present in a sample that contains 2.46 × 10^{24} atoms of hydrogen?
Molar Mass
Because we are not able to count individual atoms, it is important to have a way to convert between amounts, which are expressed in moles, and a unit of quantity that we can more easily measure, such as mass. We begin by looking at the periodic table, which tells us the relative masses of various elements.
Molar Masses of Elements
As you learned previously, the atomic masses found on the periodic table are in atomic mass units. For example, one atom of the most abundant isotope of hydrogen has a mass of approximately 1 amu, and one atom of helium has a mass of about 4 amu. Atomic masses are relative masses; they are based on the definition that one amu is equal to 1/12^{th} of the mass of a single atom of carbon12. Therefore, one atom of carbon12 has a mass of 12 amu, which is three times heavier than an atom of helium. This ratio would hold for any number of carbon and helium atoms. One hundred carbon12 atoms would have three times the mass of one hundred helium atoms. By extension, 12.00 g of carbon12 would contain the same number of atoms as 4.00 g of helium.
The relative scale of atomic masses in amu is also a relative scale of masses in grams. An alternative definition of the mole is that it is the amount of a substance that contains as many representative particles as the number of atoms in exactly 12 g of carbon12. In other words, exactly 12 g of carbon12 contains one mole, or 6.02 × 10^{23} atoms of carbon12. Likewise, 4.00 g of helium also contains one mole, or 6.02 × 10^{23} atoms of helium. The atomic mass of an element, expressed in grams, is the mass of one mole of that element. Molar mass is defined as the mass of one mole of representative particles of a substance. By looking at the periodic table, we can see that the molar mass of lithium is 6.94, the molar mass of zinc is 65.38, and the molar mass of gold is 196.97. Each of these quantities contains 6.02 × 10^{23} atoms of that particular element. The units for molar mass are grams per mole (g/mol). For example, the molar mass of zinc is 65.38 g/mol.
Recall that the atomic masses on the periodic table are generally not whole numbers because each atomic mass is a weighted average of all the naturally occurring isotopes of that element. Since any usable quantity of an element contains a very, very large number of atoms, those weighted averages in grams can be used as the molar mass of the element. For our purposes, we will use the molar masses rounded to the hundredths place (two digits after the decimal point).
Molar Masses of Compounds
The molecular formula of carbon dioxide is CO_{2}. One molecule of carbon dioxide consists of 1 atom of carbon and 2 atoms of oxygen. We can calculate the mass of one molecule of carbon dioxide by adding together the masses of 1 atom of carbon and 2 atoms of oxygen.


 12.01 amu + 2(16.00 amu) = 44.01 amu

The molecular mass of a compound is the mass of one molecule of that compound. The molecular mass of carbon dioxide is 44.01 amu.
Recall that ionic compounds do not exist as discrete molecules, but rather as an extended threedimensional network of ions called a crystal lattice. The empirical formula of an ionic compound tells us the ratio of the ions in the crystal. The mass of one formula unit of an ionic compound is called the formula mass. The formula mass of sodium sulfide, Na_{2}S, can be calculated as follows:


 2(22.99 amu) + 32.06 amu = 78.04 amu

The formula mass is the sum of the masses of all the atoms represented in a chemical formula. The term formula mass is applicable to molecular compounds, ionic compounds, or ions.
The molar mass of any compound is the mass in grams of one mole of that compound. One mole of carbon dioxide molecules has a mass of 44.01 g, while one mole of sodium sulfide formula units has a mass of 78.04 g. Their molar masses are 44.01 g/mol and 78.04 g/mol, respectively. In both cases, that is the mass of 6.02 × 10^{23} representative particles. The representative particle of CO_{2} is the molecule, while for Na_{2}S, it is the formula unit.
Sample Problem 10.3: Molar Mass of a Compound
Calcium nitrate, Ca(NO_{3})_{2}, is used as a component in fertilizer. Determine the molar mass of calcium nitrate.
Step 1: List the known and unknown quantities and plan the problem.
Known
 formula = Ca(NO_{3})_{2}
 molar mass of Ca = 40.08 g/mol
 molar mass of N = 14.01 g/mol
 molar mass of O = 16.00 g/mol
Unknown
 molar mass of Ca(NO_{3})_{2}
First, we need to analyze the formula. Since Ca lacks a subscript, there is one Ca atom per formula unit. The 2 outside the parentheses means that there are two nitrate ions per formula unit, and each nitrate ion consists of one nitrogen atom and three oxygen atoms. Therefore, there are a total of 1 × 2 = 2 nitrogen atoms and 3 × 2 = 6 oxygen atoms per formula unit. Thus, 1 mol of calcium nitrate contains 1 mol of Ca atoms, 2 mol of N atoms, and 6 mol of O atoms.
Step 2: Calculate
Use the molar mass of each atom together with the quantity of each atom in the formula to find the total molar mass.
\begin{align*}
& 1 \ \text{mol Ca} \times \frac{40.08 \ \text{g Ca}}{1 \ \text{mol Ca}}=40.08 \ \text{g Ca} \\
& 2 \ \text{mol N} \times \frac{14.01 \ \text{g N}}{1 \ \text{mol N}}=28.02 \ \text{g N} \\
& 6 \ \text{mol O} \times \frac{16.00 \ \text{g O}}{1 \ \text{mol O}}=96.00 \ \text{g O}
\end{align*}
molar mass of Ca(NO_{3})_{2} = 40.08 g + 28.02 g + 96.00 g = 164.10 g/mol
Step 3: Think about your result.
The molar mass is the mass in grams of 1 mol of calcium nitrate. It is expressed to the hundredths place because the numbers being added together are expressed to the hundredths place.
 Calculate the molar masses of the following compounds.
 C_{2}H_{6}
 (NH_{4})_{2}SO_{4}
Lesson Summary
 The amount of matter in a given sample can be measured by its mass, volume, or the number of particles.
 A mole of any substance contains Avogadro’s number (6.02 × 10^{23}) representative particles of the substance. A representative particle can be an atom, an ion, a molecule, or a formula unit.
 The molar mass of an element is its atomic mass expressed in grams and is equal to the mass of one mole of atoms of that element.
 The molar mass of a compound is the mass of one mole of representative particles of the compound. Molar mass is calculated by multiplying the molar mass of each element in the compound by the number of atoms of that element present in one formula unit and adding the resulting values together.
Lesson Review Questions
Reviewing Concepts
 What is Avogadro’s number and what does it represent?
 What is the representative particle for each of the following substances?
 barium chloride
 silicon
 nitrogen gas
 water
 How many oxygen atoms are there in a representative particle of each of the following substances?
 KClO_{4}
 CH_{3}COOH
 Al(NO_{3})_{3}
 What is wrong with the following statements?
 A mole of any substance contains the same number of atoms.
 One mole of water contains Avogadro’s number of atoms.
 If the atomic mass of a certain element is 69.72 amu, what is its molar mass?
Problems
 Calculate the number of representative particles in each of the following.
 0.0391 mol Ne
 3.72 mol NH_{3}
 8.00 mol CaF_{2}
 1.35 × 10^{−4} mol Pb^{2+}
 Calculate the number of moles represented by each of the following quantities.
 3.11 × 10^{24} molecules of NO_{2}
 8.06 × 10^{21} atoms of Pt
 How many iodine atoms are in 2.45 mol of BaI_{2}?
 Calculate the molar masses of the following substances.
 PCl_{3}
 BaCO_{3}
 Fe_{2}(SO_{4})_{3}
 Pb(CH_{3}COO)_{2}
 3.50 moles of a certain compound contains 1.05 × 10^{25} carbon atoms. How many carbon atoms are in the formula of this compound?
Further Reading / Supplemental Links
 Kieffer, W.F., Mole Concept in Chemistry, Van Nostrand Reinhold Company, 1973.
 National Mole Day Foundation, INC., (http://www.moleday.org/)
 Molar Mass Calculator, (http://www.webpc.org/mmcalc.php)
Points to Consider
The molar mass of a compound can be used to convert between the mass of a substance (in grams) and its amount (in moles).
 What would the conversion factors for these calculations look like?
 Is there a way to convert between moles and the volume of a substance?
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