6.1 The Mole LEA R N IN G O BJEC TIV E 1. Define the mole unit. Figure 6.1 "Water Molecules" shows that we need 2 hydrogen atoms and 1 oxygen
atom to make 1 water molecule. If we want to make 2 water molecules, we will
need 4 hydrogen atoms and 2 oxygen atoms. If we want to make 5 molecules of
water, we need 10 hydrogen atoms and 5 oxygen atoms. The ratio of atoms we
will need to make any number of water molecules is the same: 2 hydrogen atoms
to 1 oxygen atom.
Figure 6.1 Water Molecules
The ratio of hydrogen atoms to oxygen atoms used to make water molecules is
always 2:1, no matter how many water molecules are being made.
One problem we have, however, is that it is extremely difficult, if not impossible,
to organize atoms one at a time. As stated in the introduction, we deal with
billions of atoms at a time. How can we keep track of so many atoms (and
molecules) at a time? We do it by using mass rather than by counting individual
atoms.
A hydrogen atom has a mass of approximately 1 u. An oxygen atom has a mass of
approximately 16 u. The ratio of the mass of an oxygen atom to the mass of a
hydrogen atom is therefore approximately 16:1.
If we have 2 atoms of each element, the ratio of their masses is approximately
32:2, which reduces to 16:1—the same ratio. If we have 12 atoms of each element,
the ratio of their total masses is approximately (12 × 16):(12 × 1), or 192:12, which
also reduces to 16:1. If we have 100 atoms of each element, the ratio of the masses
is approximately 1,600:100, which again reduces to 16:1. As long as we have equal
numbers of hydrogen and oxygen atoms, the ratio of the masses will always be
16:1.
The same consistency is seen when ratios of the masses of other elements are
compared. For example, the ratio of the masses of silicon atoms to equal numbers
of hydrogen atoms is always approximately 28:1, while the ratio of the masses of
calcium atoms to equal numbers of lithium atoms is approximately 40:7.
So we have established that the masses of atoms are constant with respect to each
other, as long as we have the same number of each type of atom. Consider a more
macroscopic example. If a sample contains 40 g of Ca, this sample has the same
number of atoms as there are in a sample of 7 g of Li. What we need, then, is a
number that represents a convenient quantity of atoms so we can relate
macroscopic quantities of substances. Clearly even 12 atoms are too few because
atoms themselves are so small. We need a number that represents billions and
billions of atoms.
Chemists use the term mole to represent a large number of atoms or molecules.
Just as a dozen implies 12 things, a mole (mol) represents 6.022 × 1023 things.
The number 6.022 × 1023, called Avogadro’s number after the 19th-century
chemist Amedeo Avogadro, is the number we use in chemistry to represent
macroscopic amounts of atoms and molecules. Thus, if we have 6.022 × 1023 O
atoms, we say we have 1 mol of O atoms. If we have 2 mol of Na atoms, we have 2
× (6.022 × 1023) Na atoms, or 1.2044 × 1024 Na atoms. Similarly, if we have 0.5
mol of benzene (C6H6) molecules, we have 0.5 × (6.022 × 1023) C6H6 molecules,
or 3.011 × 1023 C6H6 molecules.
Note A mole represents a very large number! If 1 mol of quarters were stacked in a
column, it could stretch back and forth between Earth and the sun 6.8 billion
times.
Notice that we are applying the mole unit to different types of chemical entities.
In these examples, we cited moles of atoms and moles of molecules. The
word mole represents a number of things—6.022 × 1023 of them—but does not by
itself specify what “they” are. They can be atoms, formula units (of ionic
compounds), or molecules. That information still needs to be specified.
Because 1 H2 molecule contains 2 H atoms, 1 mol of H2 molecules (6.022 ×
1023molecules) has 2 mol of H atoms. Using formulas to indicate how many
atoms of each element we have in a substance, we can relate the number of moles
of molecules to the number of moles of atoms. For example, in 1 mol of ethanol
(C2H6O), we can construct the following relationships (Table 6.1 "Molecular
Relationships"):
Table 6.1 Molecular Relationships
The following example illustrates how we can use these relationships as
conversion factors.
EX A M P LE 1 If a sample consists of 2.5 mol of ethanol (C2H6O), how many moles of carbon atoms, hydrogen atoms, and oxygen atoms does it have? Solution Using the relationships in Table 6.1 "Molecular Relationships", we apply the appropriate conversion factor for each element: Note how the unit mol C2H6O molecules cancels algebraically. Similar equations can be constructed for determining the number of H and O atoms: SK ILL-­‐BU ILD IN G EXER C ISE 1. If a sample contains 6.75 mol of Na2SO4, how many moles of sodium atoms, sulfur atoms, and oxygen atoms does it have? The fact that 1 mol equals 6.022 × 1023 items can also be used as a conversion
factor.
EX A M P LE 2 How many formula units are present in 2.34 mol of NaCl? How many ions are in 2.34 mol? Solution Typically in a problem like this, we start with what we are given and apply the appropriate conversion factor. Here, we are given a quantity of 2.34 mol of NaCl, to which we can apply the definition of a mole as a conversion factor: Because there are two ions per formula unit, there are in the sample. SK ILL-­‐BU ILD IN G EXER C ISE 1. How many molecules are present in 16.02 mol of C4H10? How many atoms are in 16.02 mol? C O N C EP T R EV IEW EX ER CISE 1. What is a mole? A N SW ER 2. A mole is 6.022 × 1023 things. K EY T A K EA W A Y  A mole is 6.022 × 1023 things. EX ER C ISES 1. How many dozens are in 1 mol? Express your answer in proper scientific notation. 2. A gross is a dozen dozen, or 144 things. How many gross are in 1 mol? Express your answer in proper scientific notation. 3. How many moles of each type of atom are in 1.0 mol of C6H12O6? 4. How many moles of each type of atom are in 1.0 mol of K2Cr2O7? 5. How many moles of each type of atom are in 2.58 mol of Na2SO4? 6. How many moles of each type of atom are in 0.683 mol of C34H32FeN4O4? (This is the formula of heme, a component of hemoglobin.) 7. How many molecules are in 16.8 mol of H2O? 8. How many formula units are in 0.778 mol of iron(III) nitrate? 9. A sample of gold contains 7.02 × 1024 atoms. How many moles of gold is this? 10. A flask of mercury contains 3.77 × 1022 atoms. How many moles of mercury are in the flask? 11. An intravenous solution of normal saline may contain 1.72 mol of sodium chloride (NaCl). How many sodium and chlorine atoms are present in the solution? 12. A lethal dose of arsenic is 1.00 × 1021 atoms. How many moles of arsenic is this? A N SW ER S 1. 5.018 × 1022 dozens 3. 6.0 mol of C atoms, 12.0 mol of H atoms, and 6.0 mol of O atoms 5. 5.16 mol of Na atoms, 2.58 mol of S atoms, and 10.32 mol of O atoms 7. 1.012 × 1025 molecules 9. 11.7 mol 11. 1.04 × 1024 Na atoms and 1.04 × 1024 Cl atoms 6.2 Atomic and Molar Masses LEA R N IN G O BJEC TIV E 1. Learn how the masses of moles of atoms and molecules are expressed. Now that we have introduced the mole and practiced using it as a conversion
factor, we ask the obvious question: why is the mole that particular number of
things? Why is it 6.022 × 1023 and not 1 × 1023 or even 1 × 1020?
The number in a mole, Avogadro’s number, is related to the relative sizes of the
atomic mass unit and gram mass units. Whereas one hydrogen atom has a mass
of approximately 1 u, 1 mol of H atoms has a mass of approximately 1 gram. And
whereas one sodium atom has an approximate mass of 23 u, 1 mol of Na atoms
has an approximate mass of 23 grams.
One mole of a substance has the same mass in grams that one atom or molecule
has in atomic mass units. The numbers in the periodic table that we identified as
the atomic masses of the atoms not only tell us the mass of one atom in u but also
tell us the mass of 1 mol of atoms in grams.
Note One mole of a substance has the same mass in grams that one atom or molecule
has in atomic mass units.
EX A M P LE 3 What is the mass of each quantity? 1. 1 mol of Al atoms 2. 2 mol of U atoms Solution 1. One mole of Al atoms has a mass in grams that is numerically equivalent to the atomic mass of aluminum. The periodic table shows that the atomic mass (rounded to two decimal points) of Al is 26.98, so 1 mol of Al atoms has a mass of 26.98 g. 2. According to the periodic table, 1 mol of U has a mass of 238.03 g, so the mass of 2 mol is twice that, or 476.06 g. SK ILL-­‐BU ILD IN G EXER C ISE What is the mass of each quantity? 1. 1 mol of Au atoms 2. 5 mol of Br atoms The mole concept can be extended to masses of formula units and molecules as
well. The mass of 1 mol of molecules (or formula units) in grams is numerically
equivalent to the mass of one molecule (or formula unit) in atomic mass units.
For example, a single molecule of O2 has a mass of 32.00 u, and 1 mol of
O2 molecules has a mass of 32.00 g. As with atomic mass unit–based masses, to
obtain the mass of 1 mol of a substance, we simply sum the masses of the
individual atoms in the formula of that substance. The mass of 1 mol of a
substance is referred to as its molar mass, whether the substance is an element,
an ionic compound, or a covalent compound.
EX A M P LE 4 What is the mass of 1 mol of each substance? 1. NaCl 2. bilirubin (C33H36N4O6), the principal pigment present in bile (a liver secretion) Solution 1. Summing the molar masses of the atoms in the NaCl formula unit gives The mass of 1 mol of NaCl is 58.45 g. 2. Multiplying the molar mass of each atom by the number of atoms of that type in bilirubin’s formula and adding the results, we get The mass of 1 mol of bilirubin is 584.69 g. SK ILL-­‐BU ILD IN G EXER C ISE What is the mass of 1 mol of each substance? 1. barium sulfate (BaSO4), used to take X rays of the gastrointestional tract 2. adenosine (C10H13N5O4), a component of cell nuclei crucial for cell division Be careful when counting atoms. In formulas with polyatomic ions in
parentheses, the subscript outside the parentheses is applied to every atom inside
the parentheses. For example, the molar mass of Ba(OH)2 requires the sum of 1
mass of Ba, 2 masses of O, and 2 masses of H:
1 Ba molar mass: 1 × 137.33 g = 137.33 g
2 O molar mass:
2 × 16.00 g =
32.00 g
2 H molar mass:
2 × 1.01 g =
2.02 g
Total:
171.35 g
Because molar mass is defined as the mass for 1 mol of a substance, we can refer
to molar mass as grams per mole (g/mol). The division sign (/) implies “per,” and
“1” is implied in the denominator. Thus, the molar mass of bilirubin can be
expressed as 584.05 g/mol, which is read as “five hundred eighty four point zero
five grams per mole.”
C O N C EP T R EV IEW EX ER CISES 1. How are molar masses of the elements determined? 2. How are molar masses of compounds determined? A N SW ER S 1. Molar masses of the elements are the same numeric value as the masses of a single atom in atomic mass units but in units of grams instead. 2. Molar masses of compounds are calculated by adding the molar masses of their atoms. K EY T A K EA W A Y  The mass of moles of atoms and molecules is expressed in units of grams. EX ER C ISES 1. What is the molar mass of Si? What is the molar mass of U? 2. What is the molar mass of Mn? What is the molar mass of Mg? 3. What is the molar mass of FeCl2? What is the molar mass of FeCl3? 4. What is the molar mass of C6H6? What is the molar mass of C6H5CH3? 5 What is the molar mass of (NH4)2S? What is the molar mass of Ca(OH)2? 6. What is the molar mass of (NH4)3PO4? What is the molar mass of Sr(HCO3)2? 7. Aspirin (C9H8O4) is an analgesic (painkiller) and antipyretic (fever reducer). What is the molar mass of aspirin? 8. Ibuprofen (C13H18O2) is an analgesic (painkiller). What is the molar mass of ibuprofen? 9. Morphine (C17H19NO3) is a narcotic painkiller. What is the mass of 1 mol of morphine? 10. Heroin (C21H23NO5) is a narcotic drug that is a derivative of morphine. What is the mass of 1 mol of heroin? A N SW ER S 1. 28.09 g/mol; 238.00 g/mol 3. 126.75 g/mol; 162.20 g/mol 5. 68.15 g/mol; 74.10 g/mol 7. 180.17 g/mol 9. 285.36 g 6.3 Mole-­‐Mass Conversions LEA R N IN G O BJEC TIV E 1. Convert quantities between mass units and mole units. Example 3 in Section 6.2 "Atomic and Molar Masses" stated that the mass of 2
mol of U is twice the molar mass of uranium. Such a straightforward exercise
does not require any formal mathematical treatment. Many questions concerning
mass are not so straightforward, however, and require some mathematical
manipulations.
The simplest type of manipulation using molar mass as a conversion factor is
a mole-mass conversion (or its reverse, a mass-mole conversion). In such a
conversion, we use the molar mass of a substance as a conversion factor to
convert mole units into mass units (or, conversely, mass units into mole units).
We established that 1 mol of Al has a mass of 26.98 g (Example 3 in Section 6.2
"Atomic and Molar Masses"). Stated mathematically,
1 mol Al = 26.98 g Al
We can divide both sides of this expression by either side to get one of two
possible conversion factors:
The first conversion factor can be used to convert from mass to moles, and the
second converts from moles to mass. Both can be used to solve problems that
would be hard to do “by eye.”
Note The algebra skills we are using here are the same skills that we used in Chapter 1
"Chemistry, Matter, and Measurement" to perform unit conversions.
EX A M P LE 5 What is the mass of 3.987 mol of Al? Solution The first step in a conversion problem is to decide what conversion factor to use. Because we are starting with mole units, we want a conversion factor that will cancel the mole unit and introduce the unit for mass in the numerator. Therefore, we should use the conversion factor. We start with the given quantity and multiply by the conversion factor: Note that the mol units cancel algebraically. (The quantity 3.987 mol is understood to be in the numerator of a fraction that has 1 in the unwritten denominator.) Canceling and solving gives Our final answer is expressed to four significant figures. SK ILL-­‐BU ILD IN G EXER C ISE 1. How many moles are present in 100.0 g of Al? (Hint: you will have to use the other conversion factor we obtained for aluminum.) Conversions like this are possible for any substance, as long as the proper atomic
mass, formula mass, or molar mass is known (or can be determined) and
expressed in grams per mole. Figure 6.2 "A Simple Flowchart for Converting
between Mass and Moles of a Substance" is a chart for determining what
conversion factor is needed, andFigure 6.3 "A Flowchart Illustrating the Steps in
Performing a Unit Conversion" is a flow diagram for the steps needed to perform
a conversion.
Figure 6.2 A Simple Flowchart for Converting between Mass and Moles of a Substance
It takes one mathematical step to convert from moles to mass or from mass to
moles.
Figure 6.3 A Flowchart Illustrating the Steps in Performing a Unit Conversion
When performing many unit conversions, the same logical steps can be taken.
EX A M P LE 6 A biochemist needs 0.00655 mol of bilirubin (C33H36N4O6) for an experiment. How many grams of bilirubin will that be? Solution To convert from moles to mass, we need the molar mass of bilirubin, which we can determine from its chemical formula: The molar mass of bilirubin is 584.69 g. (We did this calculation in Example 4 inSection 6.2 "Atomic and Molar Masses".) Using the relationship 1 mol bilirubin = 584.69 g bilirubin we can construct the appropriate conversion factor for determining how many grams there are in 0.00655 mol. Following the steps from Figure 6.3 "A Flowchart Illustrating the Steps in Performing a Unit Conversion": The mol bilirubin unit cancels. The biochemist needs 3.83 g of bilirubin. SK ILL-­‐BU ILD IN G EXER C ISE 1. A chemist needs 457.8 g of KMnO4 to make a solution. How many moles of KMnO4 is that? C O N C EP T R EV IEW EX ER CISES 1. What relationship is needed to perform mole-­‐mass conversions? 2. What information determines which conversion factor is used in a mole-­‐mass conversion? A N SW ER S 1. The atomic or molar mass is needed for a mole-­‐mass conversion. 2. The unit of the initial quantity determines which conversion factor is used. K EY T A K EA W A Y  It is possible to convert between moles of material and mass of material. EX ER C ISES 1. What is the mass of 8.603 mol of Fe metal? 2. What is the mass of 0.552 mol of Ag metal? 3. What is the mass of 6.24 × 104 mol of Cl2 gas? 4. What is the mass of 0.661 mol of O2 gas? 5 What is the mass of 20.77 mol of CaCO3? 6 What is the mass of 9.02 × 10−3 mol of the hormone epinephrine (C9H13NO3)? 7 How many moles are present in 977.4 g of NaHCO3? 8. How many moles of erythromycin (C37H67NO13), a widely used antibiotic, are in 1.00 × 103 g of the substance? 9. Cortisone (C21H28O5) is a synthetic steroid that is used as an anti-­‐inflammatory drug. How many moles of cortisone are present in one 10.0 mg tablet? 10. Recent research suggests that the daily ingestion of 85 mg of aspirin (also known as acetylsalicylic acid, C9H8O4) will reduce a person’s risk of heart disease. How many moles of aspirin is that? A N SW ER S 1. 480.5 g 3. 4.42 × 106 g 5. 2,079 g 7. 11.63 mol 9. 2.77 × 10−5 mol This text was adapted by The Saylor Foundation under a Creative Commons Attribution-­‐NonCommercial-­‐ShareAlike 3.0 License without attribution as requested by the work’s original creator or licensee.