# 2.8: Atoms and the Mole

- Page ID
- 158410

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- Define the mole
- Differentiate molar mass from molecular weight
- Calculate molar mass of a compound from its formula

## The Mole

Dalton’s theory that each chemical compound has a particular combination of atoms and that the ratios of the numbers of atoms of the elements present are usually small whole numbers. It also describes the law of multiple proportions, which states that the ratios of the masses of elements that form a series of compounds are small whole numbers. The problem for Dalton and other early chemists was to discover the quantitative relationship between the number of atoms in a chemical substance and its mass. Because the masses of individual atoms are so minuscule (on the order of 10^{−23} g/atom), chemists do not measure the mass of individual atoms or molecules. In the laboratory, for example, the masses of compounds and elements used by chemists typically range from milligrams to grams, while in industry, chemicals are bought and sold in kilograms and tons. To analyze the transformations that occur between individual atoms or molecules in a chemical reaction, it is therefore essential for chemists to know how many atoms or molecules are contained in a measurable quantity in the laboratory—a given mass of sample. The unit that provides this link is the mole (mol), from the Latin *moles*, meaning “pile” or “heap.”

Many familiar items are sold in numerical quantities with distinct names. For example, cans of soda come in a six-pack, eggs are sold by the dozen (12), and pencils often come in a gross (12 dozen, or 144). Sheets of printer paper are packaged in reams of 500, a seemingly large number. Atoms are so small, however, that even 500 atoms are too small to see or measure by most common techniques. Any readily measurable mass of an element or compound contains an extraordinarily large number of atoms, molecules, or ions, so an extremely large numerical unit is needed to count them. The mole is used for this purpose.

A mole is defined as the amount of a substance that contains the number of carbon atoms in exactly 12 g of isotopically pure carbon-12. According to the most recent experimental measurements, this mass of carbon-12 contains 6.022142 × 10^{23} atoms, but for most purposes 6.022 × 10^{23} provides an adequate number of significant figures. Just as 1 mole of atoms contains 6.022 × 10^{23} atoms, 1 mole of eggs contains 6.022 × 10^{23} eggs. This number is called **Avogadro’s number**, after the 19th-century Italian scientist who first proposed a relationship between the volumes of gases and the numbers of particles they contain.

It is not obvious why eggs come in dozens rather than 10s or 14s, or why a ream of paper contains 500 sheets rather than 400 or 600. The definition of a mole—that is, the decision to base it on 12 g of carbon-12—is also arbitrary. The important point is that 1 mole of carbon—or of anything else, whether atoms, compact discs, or houses—always has the same number of objects: 6.022 × 10^{23}.

One mole always has the same number of objects: 6.022 × 1023.

To appreciate the magnitude of Avogadro’s number, consider a mole of pennies. Stacked vertically, a mole of pennies would be 4.5 × 10^{17} mi high, or almost six times the diameter of the Milky Way galaxy. If a mole of pennies were distributed equally among the entire population on Earth, each person would have more than one trillion dollars. The mole is so large that it is useful only for measuring very small objects, such as atoms.

The concept of the mole allows scientists to count a specific number of individual atoms and molecules by weighing measurable quantities of elements and compounds. To obtain 1 mol of carbon-12 atoms, one weighs out 12 g of isotopically pure carbon-12. Because each element has a different atomic mass, however, a mole of each element has a different mass, even though it contains the same number of atoms (6.022 × 10^{23}). This is analogous to the fact that a dozen extra large eggs weighs more than a dozen small eggs, or that the total weight of 50 adult humans is greater than the total weight of 50 children. Because of the way the mole is defined, for every element the number of grams in a mole is the same as the number of atomic mass units in the atomic mass of the element. For example, the mass of 1 mol of magnesium (atomic mass = 24.305 amu) is 24.305 g. Because the atomic mass of magnesium (24.305 amu) is slightly more than twice that of a carbon-12 atom (12 amu), the mass of 1 mol of magnesium atoms (24.305 g) is slightly more than twice that of 1 mol of carbon-12 (12 g). Similarly, the mass of 1 mol of helium (atomic mass = 4.002602 amu) is 4.002602 g, which is about one-third that of 1 mol of carbon-12. Using the concept of the mole, Dalton’s theory can be restated: 1 mol of a compound is formed by combining elements in amounts whose mole ratios are small whole numbers. For example, 1 mol of water (H_{2}O) has 2 mol of hydrogen atoms and 1 mol of oxygen atoms.

Molar Mass

The molar mass of a substance is defined as the mass in grams of 1 mole of that substance. One mole of isotopically pure carbon-12 has a mass of 12 g. For an element, the molar mass is the mass of 1 mol of atoms of that element; for a covalent molecular compound, it is the mass of 1 mol of molecules of that compound; for an ionic compound, it is the mass of 1 mol of formula units. That is, the molar mass of a substance is the mass (in grams per mole) of 6.022 × 10^{23} atoms, molecules, or formula units of that substance. In each case, the number of grams in 1 mol is the same as the number of atomic mass units that describe the atomic mass, the molecular mass, or the formula mass, respectively.

The molar mass of any substance is its atomic mass, molecular mass, or formula mass in grams per mole.

The periodic table lists the atomic mass of carbon as 12.011 amu; the average molar mass of carbon—the mass of 6.022 × 10^{23} carbon atoms—is therefore 12.011 g/mol:

Substance (formula) |
Atomic, Molecular, or Formula Mass (amu) |
Molar Mass (g/mol) |
---|---|---|

carbon (C) | 12.011 (atomic mass) | 12.011 |

ethanol (C_{2}H_{5}OH) |
46.069 (molecular mass) | 46.069 |

calcium phosphate [Ca_{3}(PO_{4})_{2}] |
310.177 (formula mass) | 310.177 |

The molar mass of naturally-occurring carbon is different from that of carbon-12, and is not an integer because carbon occurs as a mixture of carbon-12, carbon-13, and carbon-14. One mole of carbon still has 6.022 × 10^{23} carbon atoms, but 98.89% of those atoms are carbon-12, 1.11% are carbon-13, and a trace (about 1 atom in 1012) are carbon-14. Similarly, the molar mass of uranium is 238.03 g/mol, and the molar mass of iodine is 126.90 g/mol. When dealing with elements such as iodine and sulfur, which occur as a diatomic molecule (I_{2}) and a polyatomic molecule (S_{8}), respectively, molar mass usually refers to the mass of 1 mol of atoms of the element—in this case I and S, not to the mass of 1 mol of molecules of the element (I_{2} and S_{8}).

The molar mass of ethanol is the mass of ethanol (C_{2}H_{5}OH) that contains 6.022 × 10^{23} ethanol molecules. The molecular mass of ethanol is 46.069 amu, that is, the mass of 1 molecule of ethanol. But, we are interested in the molar mass of ethanol, the number of grams per mole of ethanol. Because 1 mol of ethanol contains 2 mol of carbon atoms (2 × 12.011 g), 6 mol of hydrogen atoms (6 × 1.0079 g), and 1 mol of oxygen atoms (1 × 15.9994 g), its molar mass is 46.069 g/mol.

Similarly, the formula mass of calcium phosphate [Ca_{3}(PO_{4})_{2}] is 310.177 amu, so its molar mass is 310.177 g/mol. This is the mass of calcium phosphate that contains 6.022 × 10^{23} formula units.

Exercise \(\PageIndex{1}\)

Which has a greater number of particles?Which of the following has a greater mass?

A) 2.5 moles of He

B) 2.5 moles of Ar

**Answer**-
Both have the same number of particles because there are 2.5 moles of each. That is similar to saying 2.5 dozen of donuts is the same number of items as the 2.5 dozen of cars. However, 2.5 moles of cars is going to have a larger mass than 2.5 dozen of trucks. By the same token, 2.5 moles of Ar is going to have more mass than 2.5 moles of He, even if they have the same number of particles.

### Practice

Calculate the molar mass of the following.

1. Aluminum Sulfate

2. Benzene (C_{6}H_{6})

3. Acetylene (C_{2}H_{2})

4. Water

5. Hydrogen Peroxide

6. Sodium Bicarbonate

7. Ammonium Nitrate

8. Potassium Perchlorate

9. Silver Chloride

10. Acetic Acid

### Answers

1. Aluminum Sulfate (Al_{2}(SO_{4})_{3}) = **342.17g/mol**

2. Benzene (C_{6}H_{6}) = **78.12g/mol**

3. Acetylene (C_{2}H_{2}) = **26.04g****/mol**

4. Water (H_{2}O) = **18.02g****/mol**

5. Hydrogen Peroxide (H_{2}O_{2}) = **34.02g****/mol**

6. Sodium Bicarbonate (NaHCO_{3}) = **84.02g****/mol**

7. Ammonium Nitrate (NH_{4}NO_{3}) = **80.08g****/mol**

8. Potassium Perchlorate (ClKO_{4}) = **138.54g****/mol**

9. Silver Chloride (AgCl) = **143.32g****/mol**

10. Acetic Acid (CH_{3}COOH) = **60.06g****/mol**

#### Contributors

- Bob Belford (UALR) and November Palmer (UALR)
- Ronia Kattoum (UA of Little Rock)
- Content used from: 3.4: Avogadro's Number and the Mole