How many molecules are in a sip of water?
A small sip of water is about 18 g, and it holds an almost unimaginable number of water molecules: about 6 × 10²³. Nobody can count particles that small one by one. In everyday life we count in bundles: eggs by the dozen (12), paper by the ream (500 sheets). Chemists have a bundle for particles too, called the mole. Once you know it, weighing a sample tells you how many particles it contains.
The mole
The mole摩尔 (symbol mol) is the SI unit for counting particles. One mole of anything contains the same number of particles, called Avogadro’s number阿伏加德罗常数, NA.
- Since 2019, the SI has defined NA as exactly 6.02214076 × 10²³ mol⁻¹. It is a defined value with no uncertainty. The old definition, “the number of atoms in exactly 12 g of carbon-12,” gives almost the same number.
- In calculations we use 6.022 × 10²³, the value on the AP Chemistry formula sheet. All of the English lessons in Chapter 2.6 use it.
The number of particles N and the number of moles n are related by
Always say what kind of particle you mean. “One mole of hydrogen” could mean H atoms or H₂ molecules, so write 1 mol H or 1 mol H₂. Moles are for atoms, molecules, ions, electrons, and other tiny particles; a “mole of rice” is not a useful idea.
A molecule’s formula tells you how many moles of each atom one mole of it contains. For example, 1 mol H₂O contains 2 mol H and 1 mol O, or 3 mol of atoms in all.
Molar mass
The mass of one mole of a substance is its molar mass摩尔质量, M, usually in g/mol.
Lesson 1.3.5 explained that atomic masses are measured relative to 1/12 of the mass of a carbon-12 atom, and one mole of carbon-12 has a mass of about 12 g (exactly 12 g under the pre-2019 definition). So the molar mass in g/mol has the same number as the atomic or formula mass in amu:
| Substance | Atomic or formula mass | Molar mass | Mass of 1 mol |
|---|---|---|---|
| Fe | 55.85 amu | 55.85 g/mol | 55.85 g |
| H₂O | 18.02 amu | 18.02 g/mol | 18.02 g |
| NaCl | 58.44 amu | 58.44 g/mol | 58.44 g |
| CO₂ | 44.01 amu | 44.01 g/mol | 44.01 g |
This lesson uses the atomic masses on the AP periodic table, such as H 1.008, C 12.01, N 14.01, O 16.00, Na 22.99, Al 26.98, Cl 35.45, Fe 55.85, and Cu 63.55.
How the Chinese textbooks say it
Chinese textbooks give n a name of its own, 物质的量 (“amount of substance,” the official SI name), and always write it out, as in n(H₂O) = 2 mol. US textbooks rarely use “amount of substance” and simply say “number of moles” or “moles of water.” Both mean the same quantity. Chinese textbooks round NA to 6.02 × 10²³ mol⁻¹ and use whole-number atomic masses (with Cl 35.5), so their answers can differ slightly in the last digit.
Pick a substance and drag the amount to see how heavy a mole is, how many particles it holds, and how big a pile it makes:
Cube volume = mass ÷ density (room temperature, densities from the CRC Handbook), drawn to scale. Molar masses use the AP periodic table; Avogadro’s number is 6.022 × 10²³ mol⁻¹.
Example 1: count molecules, then atoms
How many water molecules are in 0.500 mol H₂O? How many hydrogen atoms?
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Molecules. Number of particles = moles × Avogadro’s number:
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Moles of H atoms. Each H₂O molecule has 2 H atoms, so 1 mol H₂O contains 2 mol H:
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H atoms.
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A common mistake. Reporting the number of molecules as the number of H atoms. Decide which particle is asked for, then use the subscripts in the formula.
Example 2: from particles to moles
How many moles is 1.20 × 10²⁴ molecules of CO₂? How many moles of oxygen atoms does it contain, and what is its mass?
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Moles. n = N ÷ NA:
Three significant figures, like the given count.
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Oxygen atoms. Each CO₂ has 2 O atoms:
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Mass. M(CO₂) = 12.01 + 2(16.00) = 44.01 g/mol:
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Keep extra digits. Use the unrounded 1.993 in the calculator and round only at the end; rounding to 1.99 first gives 87.6 g. The next lesson practices this kind of conversion.
- H₂
2 × 1.008 = 2.016 g/mol.
- O₂
Oxygen gas is diatomic: 2 × 16.00 = 32.00 g/mol.
- O (an oxygen atom)
The atomic mass of O is 16.00, so 16.00 g/mol. When you read “oxygen,” check whether it means O or O₂.
- H₂O
2(1.008) + 16.00 = 18.02 g/mol.
- CO₂
12.01 + 2(16.00) = 44.01 g/mol.
- N₂
2 × 14.01 = 28.02 g/mol.
- CO
12.01 + 16.00 = 28.01 g/mol, almost the same as N₂.
- CH₄
12.01 + 4(1.008) = 16.04 g/mol, close to but not the same as an O atom.
Common mistakes
Mistake 1: “Moles and grams measure the same thing.” Moles count particles; grams measure mass. Molar mass connects the two, but 2 mol of water and 2 g of water are very different amounts.
Mistake 2: “One mole of any substance has 6.022 × 10²³ atoms.” One mole has 6.022 × 10²³ of whatever particle you name. 1 mol H₂O has 6.022 × 10²³ molecules but 3 × 6.022 × 10²³ atoms. Always say which particle you are counting.
Mistake 3: “Molar mass has no units.” The molar mass of water is 18.02 g/mol, not 18.02 or 18.02 g. Keeping g/mol in your work lets dimensional analysis (2.1.2) check each step for you.
Quick check
About how many oxygen atoms are in 0.50 mol O₂?
0.50 mol O₂ contains 2 × 0.50 = 1.0 mol O atoms, which is 6.0 × 10²³ atoms.
Which statement is correct?
“Hydrogen” does not say whether it means H or H₂; molar mass is in g/mol, not g; g/mol is the unit of molar mass, not of the mole. Only the third statement is correct.
Summary
- A mole is 6.022 × 10²³ particles (exactly 6.02214076 × 10²³ by definition). Always name the particle: atoms, molecules, ions, or formula units.
- N = n × NA: multiply moles by Avogadro’s number to get particles, and divide to go back.
- Molar mass M is the mass of one mole in g/mol. It has the same number as the atomic or formula mass in amu.
On to AP (for later)
AP Chemistry Unit 1 opens with moles and molar mass. The AP formula sheet gives Avogadro’s number as 6.022 × 10²³ mol⁻¹, and the AP periodic table lists atomic masses to two decimal places, so expect your answers to be checked for significant figures. AP questions often ask for the number of atoms rather than molecules, so remember to multiply by the number of atoms in each molecule.