The Mole Concept | Open Textbooks

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The Mole

A mole (symbol: mol) is the SI unit for measuring the amount of a substance. One mole contains exactly 6.022 x 10^23 particles (atoms, molecules, ions, or formula units). This number is called Avogadro's number.

Why Do We Need the Mole?

Atoms and molecules are incredibly tiny - far too small to count individually. The mole provides a practical way to work with enormous numbers of particles:

Just as a "dozen" means 12 items, a "mole" means 6.022 x 10^23 items
The mole connects the microscopic world (atoms) to the macroscopic world (grams)
Chemical equations are balanced in terms of moles, allowing us to predict amounts of reactants and products

Avogadro's Number

Avogadro's number (NA) = 6.022 x 10^23 particles/mol. This is the number of particles in exactly one mole of any substance.

Visualizing Avogadro's Number

How big is 6.022 x 10^23? Consider these comparisons:

If you counted one atom per second, it would take about 19 quadrillion years to count one mole of atoms
One mole of pennies stacked would create a tower reaching from Earth to the sun and back over 400 million times
One mole of sand grains would cover the entire United States to a depth of several feet

Molar Mass

Molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol). Numerically, the molar mass in grams equals the atomic mass in atomic mass units (amu).

Substance	Atomic/Formula Mass (amu)	Molar Mass (g/mol)
Carbon (C)	12.01 amu	12.01 g/mol
Oxygen (O)	16.00 amu	16.00 g/mol
Water (H2O)	18.02 amu	18.02 g/mol
Sodium Chloride (NaCl)	58.44 amu	58.44 g/mol
Glucose (C6H12O6)	180.16 amu	180.16 g/mol

Calculating Molar Mass

To find the molar mass of a compound:

Identify all elements in the formula
Multiply each element's atomic mass by its subscript
Add all the values together

Example: Molar Mass of H2SO4

H: 2 atoms x 1.01 g/mol = 2.02 g/mol
S: 1 atom x 32.07 g/mol = 32.07 g/mol
O: 4 atoms x 16.00 g/mol = 64.00 g/mol
Total: 98.09 g/mol

The Mole Triangle

Three quantities are related through the mole concept:

Conversion	Formula	Relationship
Mass to Moles	moles = mass (g) / molar mass (g/mol)	Divide by molar mass
Moles to Mass	mass (g) = moles x molar mass (g/mol)	Multiply by molar mass
Moles to Particles	particles = moles x 6.022 x 10^23	Multiply by Avogadro's number
Particles to Moles	moles = particles / 6.022 x 10^23	Divide by Avogadro's number

Molar Volume of Gases

Molar Volume at STP

At Standard Temperature and Pressure (STP) - defined as 0C (273 K) and 1 atm - one mole of any ideal gas occupies exactly 22.4 liters. This is called the molar volume.

Gas	Moles	Volume at STP
Oxygen (O2)	1 mol	22.4 L
Nitrogen (N2)	1 mol	22.4 L
Hydrogen (H2)	1 mol	22.4 L
Any ideal gas	1 mol	22.4 L

SAT/ACT Connection

Science sections may present data about chemical quantities and ask you to convert between grams, moles, and number of particles. Know your key conversion factors: molar mass connects grams to moles, and Avogadro's number connects moles to particles.

💡 Examples

Example 1: Calculating Molar Mass

Problem: Calculate the molar mass of calcium hydroxide, Ca(OH)2.

Solution:

Step 1: Identify atoms and their quantities:

Ca: 1 atom
O: 2 atoms (from the subscript outside parentheses)
H: 2 atoms (from the subscript outside parentheses)

Step 2: Look up atomic masses and multiply:

Ca: 1 x 40.08 g/mol = 40.08 g/mol
O: 2 x 16.00 g/mol = 32.00 g/mol
H: 2 x 1.01 g/mol = 2.02 g/mol

Step 3: Add all values:

Answer: 40.08 + 32.00 + 2.02 = 74.10 g/mol

Example 2: Converting Grams to Moles

Problem: How many moles are in 50.0 g of water (H2O)?

Solution:

Step 1: Find the molar mass of H2O:

H: 2 x 1.01 = 2.02 g/mol
O: 1 x 16.00 = 16.00 g/mol
Total: 18.02 g/mol

Step 2: Use the formula: moles = mass / molar mass

Step 3: Calculate: 50.0 g / 18.02 g/mol = 2.77 mol

Answer: 2.77 moles of water

Example 3: Converting Moles to Number of Particles

Problem: How many molecules are in 0.50 moles of carbon dioxide (CO2)?

Solution:

Step 1: Use Avogadro's number: 1 mole = 6.022 x 10^23 particles

Step 2: Multiply moles by Avogadro's number:

0.50 mol x 6.022 x 10^23 molecules/mol

Step 3: Calculate: 3.01 x 10^23 molecules

Answer: 3.01 x 10^23 molecules of CO2

Example 4: Converting Particles to Grams

Problem: What is the mass of 1.50 x 10^24 atoms of iron (Fe)?

Solution:

Step 1: Convert atoms to moles:

moles = 1.50 x 10^24 atoms / 6.022 x 10^23 atoms/mol = 2.49 mol

Step 2: Find molar mass of Fe: 55.85 g/mol

Step 3: Convert moles to grams:

mass = 2.49 mol x 55.85 g/mol = 139 g

Answer: 139 grams of iron

Example 5: Using Molar Volume at STP

Problem: What volume does 3.5 moles of oxygen gas (O2) occupy at STP?

Solution:

Step 1: At STP, 1 mole of any gas = 22.4 L

Step 2: Multiply moles by molar volume:

Volume = 3.5 mol x 22.4 L/mol

Step 3: Calculate: 78.4 L

Answer: 78.4 liters at STP

✏️ Practice

1. Avogadro's number is approximately:

A) 6.022 x 10^20

B) 6.022 x 10^21

C) 6.022 x 10^22

D) 6.022 x 10^23

2. What is the molar mass of sulfuric acid (H2SO4)?

A) 49.1 g/mol

B) 98.1 g/mol

C) 80.1 g/mol

D) 34.1 g/mol

3. How many moles are in 44 g of CO2 (molar mass = 44 g/mol)?

A) 0.5 mol

B) 1.0 mol

C) 2.0 mol

D) 44 mol

4. One mole of any gas at STP occupies:

A) 1.0 L

B) 11.2 L

C) 22.4 L

D) 44.8 L

5. How many molecules are in 2.0 moles of water?

A) 3.01 x 10^23

B) 6.02 x 10^23

C) 1.20 x 10^24

D) 1.80 x 10^24

6. What is the mass of 0.25 moles of NaCl (molar mass = 58.44 g/mol)?

A) 14.6 g

B) 29.2 g

C) 58.4 g

D) 233.8 g

7. How many atoms are in one molecule of glucose (C6H12O6)?

A) 6

B) 12

C) 18

D) 24

8. The molar mass of an element is numerically equal to its:

A) Number of protons

B) Atomic number

C) Atomic mass in amu

D) Number of electrons

9. How many moles of oxygen atoms (not O2 molecules) are in 1 mole of H2O?

A) 0.5 mol

B) 1.0 mol

C) 2.0 mol

D) 3.0 mol

10. If 3.01 x 10^23 atoms of an element have a mass of 12 g, what is the molar mass of the element?

A) 6 g/mol

B) 12 g/mol

C) 24 g/mol

D) 36 g/mol

Click to reveal answers

D - Avogadro's number is 6.022 x 10^23 particles per mole.
B - H2SO4: 2(1.01) + 32.07 + 4(16.00) = 2.02 + 32.07 + 64.00 = 98.09 g/mol
B - moles = 44 g / 44 g/mol = 1.0 mol
C - At STP, one mole of any ideal gas occupies 22.4 L (molar volume).
C - 2.0 mol x 6.022 x 10^23 = 1.20 x 10^24 molecules
A - mass = 0.25 mol x 58.44 g/mol = 14.61 g
D - C6H12O6 has 6 C + 12 H + 6 O = 24 atoms per molecule
C - The molar mass in g/mol equals the atomic mass in amu.
B - Each H2O molecule has 1 oxygen atom, so 1 mol H2O contains 1 mol O atoms.
C - 3.01 x 10^23 is 0.5 mol. If 0.5 mol = 12 g, then 1 mol = 24 g/mol.

✅ Check Your Understanding

Question 1: Explain why the mole is sometimes called the "chemist's dozen." How does it serve a similar purpose?

Reveal Answer

Just as a "dozen" represents 12 items (making it easier to buy eggs or donuts without counting each one), a "mole" represents 6.022 x 10^23 particles. The mole allows chemists to work with countable, measurable amounts of substances rather than dealing with individual atoms or molecules. A dozen connects us to a convenient number of everyday objects; a mole connects us to a convenient amount of atoms/molecules that we can actually weigh on a balance. It's the bridge between the atomic scale (too small to measure directly) and the macroscopic scale (grams we can measure).

Question 2: Why is the molar mass of carbon-12 exactly 12 g/mol, while other elements have molar masses that aren't whole numbers?

Reveal Answer

The atomic mass unit (amu) was defined so that one atom of carbon-12 has a mass of exactly 12 amu. Since Avogadro's number was chosen so that the molar mass in grams numerically equals the atomic mass in amu, carbon-12 has a molar mass of exactly 12 g/mol by definition. Other elements have non-integer molar masses for two reasons: (1) Most elements exist as mixtures of isotopes with different masses, and the atomic mass is a weighted average, and (2) Even single isotopes have masses that include binding energy effects, making them not exactly whole numbers.

Question 3: A student says, "Since 1 mole of any gas occupies 22.4 L at STP, one mole of any gas must have the same mass." Is this correct? Explain.

Reveal Answer

The student is incorrect. While one mole of any ideal gas does occupy the same volume (22.4 L) at STP, different gases have different molar masses. For example, 1 mole of H2 (molar mass 2 g/mol) weighs only 2 grams, while 1 mole of O2 (molar mass 32 g/mol) weighs 32 grams. Both occupy 22.4 L at STP. The equal volume is due to the ideal gas law - at the same temperature and pressure, equal numbers of gas molecules occupy equal volumes, regardless of the molecule's size or mass.

Question 4: A compound's formula is C3H8O. If you have 3.01 x 10^23 molecules of this compound, how many total atoms do you have? Show your reasoning.

Reveal Answer

Step 1: Count atoms per molecule of C3H8O:
3 carbon + 8 hydrogen + 1 oxygen = 12 atoms per molecule

Step 2: Determine how many molecules we have:
3.01 x 10^23 molecules = 0.5 mol (half of Avogadro's number)

Step 3: Calculate total atoms:
3.01 x 10^23 molecules x 12 atoms/molecule = 3.61 x 10^24 atoms

Alternatively: 0.5 mol molecules x 12 atoms/molecule = 6 mol atoms = 6 x 6.022 x 10^23 = 3.61 x 10^24 atoms

🚀 Next Steps

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