What is a Mole?

The mole (symbol: mol) is the SI unit for the amount of substance.

Just as a "dozen" always means 12 items (like eggs or donuts), a "mole" always means a specific number of particles: Avogadro's Number.

$$ 1 \text{ mol} = 6.022 \times 10^{23} \text{ particles} $$

These particles can be atoms, molecules, ions, or any other microscopic entity.

How to Convert Between Moles and Particles

To convert between moles and particles (atoms or molecules), we use Avogadro's Number.

  • Moles → Particles: Multiply the moles by $6.022 \times 10^{23}$.
  • Particles → Moles: Divide the particles by $6.022 \times 10^{23}$.
$$ \text{Moles} \xrightarrow{\times 6.022 \times 10^{23}} \text{Particles (Atoms/Molecules)} $$
$$ \text{Particles (Atoms/Molecules)} \xrightarrow{\div 6.022 \times 10^{23}} \text{Moles} $$

Why do we use the Mole?

Atoms and molecules are incredibly small. If you were to count the atoms in a glass of water one by one, it would take you millions of years!

The mole allows chemists to: 1. Work with manageable numbers: instead of saying $602,200,000,000,000,000,000,000$ atoms, we just say "1 mole". 2. Connect the microscopic to the macroscopic: It bridges the gap between the tiny mass of a single atom and the grams we measure in a laboratory.

Solved Examples

When solving mole problems, we use conversion factors (fractions) or direct proportions (the rule of three).

Example: Converting Molecules to Moles

Question: Calculate how many moles are in $1.806 \times 10^{24}$ molecules of Carbon Dioxide ($CO_2$).

Solution:

We use Avogadro's number as a fraction. Since we want to find "moles", we put "1 mol" on top:

$$ 1.806 \times 10^{24} \text{ molecules } CO_2 \times \frac{1 \text{ mol } CO_2}{6.022 \times 10^{23} \text{ molecules } CO_2} = 3 \text{ mol } CO_2 $$
Example: Finding Atoms of an element in a compound

Question: How many atoms of Hydrogen are in $2$ moles of Methane ($CH_4$)?

Solution:

1. Each molecule of $CH_4$ contains 4 atoms of Hydrogen (the subscript).

2. Therefore, $1$ mole of $CH_4$ contains 4 moles of Hydrogen atoms.

$$ 2 \text{ mol } CH_4 \times \frac{4 \text{ mol H}}{1 \text{ mol } CH_4} \times \frac{6.022 \times 10^{23} \text{ atoms H}}{1 \text{ mol H}} $$
$$ = 8 \text{ mol H} \times 6.022 \times 10^{23} \text{ atoms/mol} = 4.817 \times 10^{24} \text{ atoms of Hydrogen} $$
Example: Atoms in a complex molecule (Glucose)

Question: How many atoms of Oxygen are in $0.5$ moles of Glucose ($C_6H_{12}O_6$)?

Solution:

$$ 0.5 \text{ mol } C_6H_{12}O_6 \times \frac{6 \text{ mol O}}{1 \text{ mol } C_6H_{12}O_6} \times \frac{6.022 \times 10^{23} \text{ atoms O}}{1 \text{ mol O}} = 1.806 \times 10^{24} \text{ atoms O} $$

Mole-to-Mole Calculations (Proportions)

In a balanced chemical equation, the coefficients (the numbers in front of the formulas) tell us the mole ratio between the substances. We can think of these as "recipes".

Example: $2 H_2 + O_2 \rightarrow 2 H_2O$ This recipe says that for every 2 moles of $H_2$, we need 1 mole of $O_2$ to produce 2 moles of $H_2O$.

We can use direct proportions (also known as the rule of three) or conversion factors to calculate how much of one substance we need if we know the amount of another.

Example: Calculating Moles Produced

Reaction: $N_2 + 3 H_2 \rightarrow 2 NH_3$

Question: How many moles of Ammonia ($NH_3$) are produced if we react $9$ moles of Hydrogen ($H_2$)?

Solution (using a Proportion):

$ \frac{3 \text{ mol } H_2}{2 \text{ mol } NH_3} = \frac{9 \text{ mol } H_2}{x \text{ mol } NH_3} $

$ x = \frac{9 \times 2}{3} = 6 \text{ mol } NH_3 $

Example: Calculating Moles Needed

Reaction: $N_2 + 3 H_2 \rightarrow 2 NH_3$

Question: How many moles of Nitrogen ($N_2$) are needed to react completely with $12$ moles of Hydrogen ($H_2$)?

Solution (using a Conversion Factor):

$ 12 \text{ mol } H_2 \times \frac{1 \text{ mol } N_2}{3 \text{ mol } H_2} = 4 \text{ mol } N_2 $

Example: Calculating Total Atoms

Question: How many total atoms are in $0.1$ moles of Water ($H_2O$)?

Solution:

1. Each molecule of $H_2O$ has 3 atoms ($2$ Hydrogen + $1$ Oxygen).

2. Moles of atoms:

$ n_{\text{atoms}} = 0.1 \text{ mol } H_2O imes 3 \text{ atoms/molecule} = 0.3 \text{ moles of atoms} $

3. Number of atoms ($N$):

$ N = 0.3 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} = 1.8066 imes 10^{23} \text{ atoms} $

Example: Large Scale Molecules to Moles

Question: Calculate how many moles are in $1.8066 imes 10^{25}$ molecules of Methane ($CH_4$).

Solution:

$ n = \frac{1.8066 \times 10^{25}}{6.022 \times 10^{23}} = 0.3 \times 10^2 = 30 \text{ moles} $

1. What is the numerical value of Avogadro's number? (1 points)

2. The mole is the SI unit for which physical quantity? (1 points)

3. Why do scientists use the mole? (1 points)

4. In the reaction 2H2 + O2 → 2H2O, what is the mole ratio of H2 to O2? (1 points)

5. Mole calculations practice: (4 points)

a ) How many moles are in $6.022 \times 10^{23}$ molecules of Water?

b ) Using 2H2 + O2 → 2H2O: If we have 10 mol of H2, how many mol of O2 are needed?

c ) The mole is to chemistry what the _____ is to eggs. (one word, plural)

d ) Using N2 + 3H2 → 2NH3: How many mol of NH3 are produced from 1.5 mol of N2?

6. Advanced Mole Conversions: (4 points)

a ) How many moles are in $1.2044 \times 10^{24}$ molecules? (Use decimal)

b ) How many moles of Oxygen atoms are in 2 moles of H2SO4?

c ) If we have 0.5 moles of O2, how many molecules is that? (Use 3.011e23 format)

d ) Using CH4 + 2O2 → CO2 + 2H2O: How many moles of O2 are needed to react with 3 moles of CH4?

7. Final Challenge: Atoms and Compounds (4 points)

a ) How many moles of atoms (total) are in 1 mole of CO2?

b ) Using 2Al + 3Cl2 → 2AlCl3: How many moles of AlCl3 are produced from 0.4 mol of Al?

c ) How many moles are in $3.011 \times 10^{22}$ molecules? (Careful with exponent)

d ) How many Oxygen atoms are in 0.25 moles of CO2? (Use 3.011e23 format)

8. Mole Ratio Challenge: (4 points)

a ) Using $2H_2 + O_2 \rightarrow 2H_2O$: How many moles of Water are produced from 10 moles of $O_2$?

b ) How many moles are in $1.8066 \times 10^{24}$ atoms of Iron?

c ) How many atoms of Hydrogen are in 1 mole of Water ($H_2O$)? (Use 1.2044e24 format)

d ) What is the total number of moles of atoms in 0.5 moles of $CH_4$?