Mole Conversions & Chemical Calculations

What is a Mole?

Imagine trying to count out individual atoms to make water. You'd need:

602,214,076,000,000,000,000,000 hydrogen atoms (that's 602 sextillion!)
301,107,038,000,000,000,000,000 oxygen atoms

That's... completely impossible. So chemists invented the mole - think of it as the "chemistry dozen."

🎯 The Mole = Chemistry's Counting Unit

1 mole = exactly 6.02 × 10²³ particles

This special number is called Avogadro's Number (N_A)

1 mole of atoms = 6.02 × 10²³ atoms
1 mole of molecules = 6.02 × 10²³ molecules
1 mole of pennies = 6.02 × 10²³ pennies (enough to cover Earth 400 km deep!)

💡 Why this weird number? It's chosen so that the mass of 1 mole of atoms (in grams) equals the atomic mass from the periodic table. For example:

1 carbon atom weighs 12 amu → 1 mole of carbon = 12 grams
1 oxygen atom weighs 16 amu → 1 mole of oxygen = 16 grams

1. The Mole Conversion Roadmap

The Three-Way Conversion Map

GRAMS

(what you can weigh)

↔ MOLES ↔

(the bridge)

PARTICLES

(atoms/molecules)

Grams → Moles: Divide by molar mass

Moles → Grams: Multiply by molar mass

Moles → Particles: Multiply by 6.02 × 10²³

Particles → Moles: Divide by 6.02 × 10²³

⚠️ Important Rule: Moles are ALWAYS in the middle! You almost never go directly from grams to particles - you must go through moles first!

2. Calculating Molar Mass

⚖️ Molar Mass = Mass of 1 Mole

Molar mass is the mass (in grams) of exactly 1 mole of a substance. Units: g/mol

For elements: Look at the periodic table! The number below the symbol is the molar mass.

Carbon (C): 12.01 g/mol
Oxygen (O): 16.00 g/mol
Hydrogen (H): 1.01 g/mol

For compounds: Add up the molar masses of all atoms in the formula.

Example 1: Molar Mass of Water (H₂O)

Formula breakdown: 2 hydrogen atoms + 1 oxygen atom

Hydrogen: 2 atoms × 1.01 g/mol = 2.02 g/mol

Oxygen: 1 atom × 16.00 g/mol = 16.00 g/mol

Total: 2.02 + 16.00 = 18.02 g/mol

This means 1 mole of water weighs 18.02 grams!

Example 2: Molar Mass of Magnesium Bicarbonate Mg(HCO₃)₂

Step 1: Break down the formula

The subscript "2" outside the parentheses means EVERYTHING inside is doubled:

1 Mg
2 H (because of the 2 outside parentheses)
2 C
6 O (3 inside × 2 outside)

Step 2: Calculate

Mg: 1 × 24.31 = 24.31 g/mol

H: 2 × 1.01 = 2.02 g/mol

C: 2 × 12.01 = 24.02 g/mol

O: 6 × 16.00 = 96.00 g/mol

Total: 146.35 g/mol

✅ Pro Tip: When calculating molar mass, write out EVERY atom count and mass. This helps you catch mistakes!

3. Converting Between Grams and Moles

The Two Key Formulas

\[\text{moles} = \frac{\text{grams}}{\text{molar mass}}\]

\[\text{grams} = \text{moles} \times \text{molar mass}\]

Example 3: Grams to Moles

Problem: How many moles are in 25.0 g of NaOH?

Step 1: Calculate molar mass of NaOH

Na: 22.99 g/mol, O: 16.00 g/mol, H: 1.01 g/mol
Molar mass = 22.99 + 16.00 + 1.01 = 40.00 g/mol

Step 2: Convert grams to moles

\[\text{moles} = \frac{25.0 \text{ g}}{40.00 \text{ g/mol}} = 0.625 \text{ mol}\]

Answer: 0.625 mol NaOH

Example 4: Moles to Grams

Problem: What is the mass of 4.25 × 10⁻⁴ mol of H₂SO₄?

Step 1: Molar mass of H₂SO₄

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

Step 2: Convert

\[\text{grams} = 4.25 \times 10^{-4} \text{ mol} \times 98.09 \text{ g/mol} = 0.0417 \text{ g}\]

Answer: 0.0417 g or 41.7 mg

4. Converting Between Moles and Particles

Using Avogadro's Number

\[1 \text{ mol} = 6.02 \times 10^{23} \text{ particles}\]

\[\text{particles} = \text{moles} \times 6.02 \times 10^{23}\]

\[\text{moles} = \frac{\text{particles}}{6.02 \times 10^{23}}\]

Example 5: Molecules in Sugar

Problem: How many molecules of sugar are in a 5-lb bag if the formula is C₁₂H₂₂O₁₁?

Step 1: Convert pounds to grams

\[5 \text{ lb} \times \frac{454 \text{ g}}{1 \text{ lb}} = 2270 \text{ g}\]

Step 2: Molar mass of C₁₂H₂₂O₁₁

C: 12 × 12.01 = 144.12
H: 22 × 1.01 = 22.22
O: 11 × 16.00 = 176.00
Total = 342.34 g/mol

Step 3: Grams → Moles

\[\frac{2270 \text{ g}}{342.34 \text{ g/mol}} = 6.63 \text{ mol}\]

Step 4: Moles → Molecules

\[6.63 \text{ mol} \times 6.02 \times 10^{23} \text{ molecules/mol} = 3.99 \times 10^{24} \text{ molecules}\]

Answer: 4.0 × 10²⁴ sugar molecules!

Example 6: Avogadro's Paper Stack

Problem: If a stack of 500 sheets of paper is 4.60 cm high, what will be the height, in meters, of a stack of Avogadro's number of sheets of paper?

Step 1: Height per sheet

\[\frac{4.60 \text{ cm}}{500 \text{ sheets}} = 0.0092 \text{ cm/sheet}\]

Step 2: Height of Avogadro's number of sheets

\[6.02 \times 10^{23} \text{ sheets} \times 0.0092 \text{ cm/sheet} = 5.54 \times 10^{21} \text{ cm}\]

Step 3: Convert to meters

\[5.54 \times 10^{21} \text{ cm} \times \frac{1 \text{ m}}{100 \text{ cm}} = 5.54 \times 10^{19} \text{ m}\]

Answer: 5.54 × 10¹⁹ m - That's about 587 light-years!

5. Counting Atoms in Molecules

🔬 Molecules vs Atoms

One molecule of H₂O contains 3 atoms (2 H + 1 O)

So 1 mole of H₂O contains:

6.02 × 10²³ molecules of H₂O
2 × 6.02 × 10²³ = 1.20 × 10²⁴ atoms of H
1 × 6.02 × 10²³ = 6.02 × 10²³ atoms of O
Total: 1.80 × 10²⁴ atoms

Example 7: Carbon Disulfide Breakdown

Problem: Exactly 1 mol of carbon disulfide contains:

a. how many carbon disulfide molecules?
b. how many carbon atoms?
c. how many sulfur atoms?
d. how many total atoms of all kinds?

Formula: CS₂ (1 carbon, 2 sulfur per molecule)

Solutions:

a. 1 mol = 6.02 × 10²³ CS₂ molecules

b. 1 C per molecule × 6.02 × 10²³ = 6.02 × 10²³ C atoms

c. 2 S per molecule × 6.02 × 10²³ = 1.20 × 10²⁴ S atoms

d. 3 total atoms per molecule × 6.02 × 10²³ = 1.81 × 10²⁴ total atoms

Example 8: Oxygen Atoms in Compounds

Problem: How many atoms of oxygen are in 5.0 g of MnO₂?

Step 1: Molar mass of MnO₂

Mn: 54.94, O: 2 × 16.00 = 32.00
Total = 86.94 g/mol

Step 2: Grams → Moles

\[\frac{5.0 \text{ g}}{86.94 \text{ g/mol}} = 0.0575 \text{ mol MnO}_2\]

Step 3: Moles MnO₂ → Moles O atoms

Each MnO₂ has 2 O atoms, so:
\[0.0575 \text{ mol MnO}_2 \times \frac{2 \text{ mol O}}{1 \text{ mol MnO}_2} = 0.115 \text{ mol O}\]

Step 4: Moles O → Atoms O

\[0.115 \text{ mol} \times 6.02 \times 10^{23} = 6.92 \times 10^{22} \text{ O atoms}\]

Answer: 6.9 × 10²² oxygen atoms

6. Your Mole Conversion Strategy

🎯 Step-by-Step Game Plan:

Identify what you have and what you need (grams? moles? particles?)
Calculate molar mass if needed (for grams ↔ moles conversions)
Map your route through MOLES - moles are ALWAYS the bridge!
Set up conversion factors - use molar mass or Avogadro's number
Watch your subscripts! - Don't forget atoms per molecule
Calculate and apply sig figs
Check if it makes sense - bigger numbers for atoms, smaller for moles

🧠 Quick Reference

Conversion	Factor to Use
Grams → Moles	Divide by molar mass
Moles → Grams	Multiply by molar mass
Moles → Particles	Multiply by 6.02 × 10²³
Particles → Moles	Divide by 6.02 × 10²³
Grams → Particles	Go through moles (2 steps!)

⚠️ Common Mistakes to Avoid:

Forgetting to calculate molar mass correctly (watch parentheses!)
Trying to skip moles and go directly from grams to particles
Mixing up molecules and atoms (H₂O has 3 atoms per molecule!)
Forgetting Avogadro's number: 6.02 × 10²³
Not checking sig figs in scientific notation