14.1

Molecular Mass, The Mole, and Avogadro’s Number

Often, we wish to compare the mass of one atom with another. To facilitate the comparison, a mass scale known as the atomic mass scale has been established. To set up this scale, a reference value (along with a unit) is chosen for one of the elements. The unit is called the atomic mass unit (symbol: u). By international agreement, the reference element is chosen to be the most abundant type or isotope* of carbon, which is called carbon-12. Its atomic mass* is defined to be exactly twelve atomic mass units, or 12 u. The relationship between the atomic mass unit and the kilogram is

The atomic masses of all the elements are listed in the periodic table, part of which is shown in Figure 14-1. The complete periodic table is given on the inside of the back cover. In general, the masses listed are average values and take into account the various isotopes of an element that exist naturally. For brevity, the unit “u”is often omitted from the table. For example, a magnesium atom (Mg) has an average atomic mass of 24.305 u, while that for the lithium atom (Li) is 6.941 u; thus, atomic magnesium is more massive than atomic lithium by a factor of . In the periodic table, the atomic mass of carbon (C) is given as 12.011 u, rather than exactly 12 u, because a small amount (about 1%) of the naturally occurring material is an isotope called carbon-13. The value of 12.011 u is an average that reflects the small contribution of carbon-13.

Figure 14-1 A portion of the periodic table showing the atomic number and atomic mass of each element. In the periodic table it is customary to omit the symbol “u”denoting the atomic mass unit.

The molecular mass of a molecule is the sum of the atomic masses of its atoms. For instance, elements hydrogen and oxygen have atomic masses of 1.007 94 u and 15.9994 u, respectively, so the molecular mass of a water molecule (H2O) is, therefore, .
Macroscopic amounts of materials contain large numbers of atoms or molecules. Even in a small volume of gas, 1 cm3, for example, the number is enormous. It is convenient to express such large numbers in terms of a single unit, the gram-mole, or simply the mole (symbol: mol). One gram-mole of a substance contains as many particles (atoms or molecules) as there are atoms in 12 grams of the isotope carbon-12. Experiment shows that 12 grams of carbon-12 contain . The number of atoms per mole is known as Avogadro’s number NA, after the Italian scientist Amedeo Avogadro (1776–1856):

Thus, the number of moles n contained in any sample is the number of particles N in the sample divided by the number of particles per mole NA (Avogadro’s number):

Although defined in terms of carbon atoms, the concept of a mole can be applied to any collection of objects by noting that one mole contains Avogadro’s number of objects. Thus, one mole of atomic sulfur contains , one mole of water contains , and one mole of golf balls contains . Just as the meter is the SI base unit for length, the mole is the SI base unit for expressing “the amount of a substance.”
The number n of moles contained in a sample can also be found from its mass. To see how, multiply and divide the right-hand side of the previous equation by the mass mparticle of a single particle, expressed in grams:

The numerator mparticleN is the mass of a particle times the number of particles in the sample, which is the mass m of the sample. The denominator mparticleNA is the mass of a particle times the number of particles per mole, which is the mass per mole, expressed in grams per mole. The mass per mole of carbon-12 is 12 g/mol, since, by definition, 12 grams of carbon-12 contain one mole of atoms. On the other hand, the mass per mole of sodium (Na) is 22.9898 g/mol for the following reason: as indicated in Figure 14-1, a sodium atom is more massive than a carbon-12 atom by the ratio of their atomic masses, . Therefore, the mass per mole of sodium is 1.915 82 times as great as that of carbon-12, or . Note that the numerical value of the mass per mole of sodium (22.9898) is the same as the numerical value of its atomic mass. This is true in general, so the mass per mole (in g/mol) of a substance has the same numerical value as the atomic or molecular mass of the substance (in atomic mass units).

Problem Solving Insight

Since one gram-mole of a substance contains Avogadro’s number of particles (atoms or molecules), the mass mparticle of a particle (in grams) can be obtained by dividing the mass per mole (in g/mol) by Avogadro’s number:

Example 1 illustrates how to use the concepts of the mole, atomic mass, and Avogadro’s number to determine the number of atoms and molecules present in two famous gemstones.

| Example 1 | | | The Hope Diamond and the Rosser Reeves Ruby

Figure 14-2a shows the Hope diamond (44.5 carats), which is almost pure carbon. Figure 14-2b shows the Rosser Reeves ruby (138 carats), which is primarily aluminum oxide (Al2O3). One carat is equivalent to a mass of 0.200 g. Determine (a) the number of carbon atoms in the diamond and (b) the number of Al2O3 molecules in the ruby.

Figure 14-2 (a) The Hope diamond surrounded by 16 smaller diamonds. (Dane A. Penland/Gem & Mineral Collection/Smithsonian Institution) (b) The Rosser Reeves ruby. (Chip Clark/Smithsonian Institution) Both gems are on display at the Smithsonian Institution in Washington, D.C.

Reasoning The number N of atoms (or molecules) in a sample is the number of moles n times the number of atoms per mole NA (Avogadro’s number); N = nNA. We can determine the number of moles by dividing the mass of the sample m by the mass per mole of the substance.
Solution

a. | The Hope diamond’s mass is . Since the average atomic mass of naturally occurring carbon is 12.011 u (see the periodic table on the inside of the back cover), the mass per mole of this substance is 12.011 g/mol. The number of moles of carbon in the Hope diamond is The number of carbon atoms in the Hope diamond is

b. | The mass of the Rosser Reeves ruby is . The molecular mass of an aluminum oxide molecule (Al2O3) is the sum of the atomic masses of its atoms, which are 26.9815 u for aluminum and 15.9994 u for oxygen (see the periodic table on the inside of the back cover):

Thus, the mass per mole of Al2O3 is 101.9612 g/mol. Calculations like those in part (a) reveal that the Rosser Reeves ruby contains 0.271 mol or .

The physics of gemstones.

Check Your Understanding 1

A gas mixture contains equal masses of the monatomic gases argon and neon . They are the only gases in the mixture. Of the total number of atoms, what percentage is neon/ (The answer is given at the end of the book.) Background: The mole, atomic mass, and Avogadro’s number are the concepts that must be used in answering this question.
For similar questions (including conceptual counterparts), consult Self-Assessment Test 14.1. This test is described at the end of Section 14.4.

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